{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Omni-Math Preprocessing"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Filter for non-proof problems."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4422"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import json\n",
    "\n",
    "with open(\"../train/omni_math.json\") as f:\n",
    "    omni = json.load(f)\n",
    "len(omni)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/opt/conda/envs/rllm/lib/python3.10/site-packages/tqdm/auto.py:21: TqdmWarning: IProgress not found. Please update jupyter and ipywidgets. See https://ipywidgets.readthedocs.io/en/stable/user_install.html\n",
      "  from .autonotebook import tqdm as notebook_tqdm\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "('A physicist encounters $2015$ atoms called usamons. Each usamon either has '\n",
      " \"one electron or zero electrons, and the physicist can't tell the \"\n",
      " \"difference.  The physicist's only tool is a diode. The physicist may connect \"\n",
      " 'the diode from any usamon $A$ to any other usamon $B$. (This connection is '\n",
      " 'directed.) When she does so, if usamon $A$ has an electron and usamon $B$ '\n",
      " 'does not, then the electron jumps from $A$ to $B$. In any other case, '\n",
      " 'nothing happens. In addition, the physicist cannot tell whether an electron '\n",
      " \"jumps during any given step.  The physicist's goal is to isolate two usamons \"\n",
      " 'that she is  sure are currently in the same state. Is there any series of '\n",
      " 'diode usage that makes this possible?')\n",
      "'\\\\text{No}'\n",
      "('The problem asks whether there exists a series of diode usage that allows '\n",
      " 'the physicist to isolate two usamons in the same state. The answer provided '\n",
      " 'is \"No.\" This indicates that the problem is asking for a proof to a yes/no '\n",
      " 'question. Therefore, the problem falls under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Let $P$ be a polynomial with integer coefficients such that $P(0)=0$ and\\n'\n",
      " '\\\\[\\\\gcd(P(0), P(1), P(2), \\\\ldots ) = 1.\\\\]\\n'\n",
      " 'Show there are infinitely many $n$ such that\\n'\n",
      " '\\\\[\\\\gcd(P(n)- P(0), P(n+1)-P(1), P(n+2)-P(2), \\\\ldots) = n.\\\\]')\n",
      "'\\\\text{infinitely many } n'\n",
      "('The problem asks to show that there are infinitely many $n$ that satisfy a '\n",
      " 'given property. This type of problem falls into Case 2, as it requires a '\n",
      " 'proof to demonstrate the existence of infinitely many such $n$. The provided '\n",
      " 'answer only states the conclusion but does not offer the reasoning or '\n",
      " 'argument needed to establish the claim.  Thus, a proof is required.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Determine whether or not there exist positive integers $ a$ and $ b$ such '\n",
      " 'that $ a$ does not divide $ b^n \\\\minus{} n$ for all positive integers $ n$.')\n",
      "'\\\\text{No}'\n",
      "('The problem asks to determine whether such integers exist, which is a yes/no '\n",
      " 'question. This makes it a Case 2 problem.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Call a sequence of positive integers $\\\\{a_n\\\\}$ good if for any distinct '\n",
      " 'positive integers $m,n$, one has \\n'\n",
      " '$$\\\\gcd(m,n) \\\\mid a_m^2 + a_n^2 \\\\text{ and } \\\\gcd(a_m,a_n) \\\\mid m^2 + '\n",
      " 'n^2.$$\\n'\n",
      " 'Call a positive integer $a$ to be $k$-good if there exists a good sequence '\n",
      " 'such that $a_k = a$. Does there exists a $k$ such that there are exactly '\n",
      " '$2019$ $k$-good positive integers?')\n",
      "'\\\\text{no}'\n",
      "('The problem asks whether there exists a $k$ such that there are exactly '\n",
      " \"$2019$ $k$-good positive integers. It's a yes/no question and the answer \"\n",
      " 'provided is \"no\". This indicates the problem requires a proof to show why '\n",
      " \"such a  $k$ doesn't exist. Thus, it falls under Case 2.\\n\"\n",
      " '\\n'\n",
      " 'Final Answer: [[2]]\\n')\n",
      "('Let $n=p_1^{a_1}p_2^{a_2}\\\\cdots p_t^{a_t}$ be the prime factorisation of '\n",
      " '$n$. Define $\\\\omega(n)=t$ and $\\\\Omega(n)=a_1+a_2+\\\\ldots+a_t$. Prove or '\n",
      " 'disprove:\\n'\n",
      " 'For any fixed positive integer $k$ and positive reals $\\\\alpha,\\\\beta$, '\n",
      " 'there exists a positive integer $n>1$ such that\\n'\n",
      " 'i) $\\\\frac{\\\\omega(n+k)}{\\\\omega(n)}>\\\\alpha$\\n'\n",
      " 'ii) $\\\\frac{\\\\Omega(n+k)}{\\\\Omega(n)}<\\\\beta$.')\n",
      "'\\\\text{True}'\n",
      "('The problem asks to prove or disprove a statement. The statement says '\n",
      " 'whether a positive integer $n$ exists that satisfies the given conditions. '\n",
      " 'This clearly falls under Case 2, since we are asked to prove the existence '\n",
      " 'of such an integer. Despite the answer being \"True\", this does not make the '\n",
      " 'problem Case 1, since the core of the problem is to provide a proof.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Determine whether or not there exist two different sets $A,B$, each '\n",
      " 'consisting of at most $2011^2$ positive integers, such that every $x$ with '\n",
      " '$0 < x < 1$ satisfies the following inequality:\\n'\n",
      " '\\\\[\\\\left| \\\\sum_{a \\\\in A} x^a - \\\\sum_{b \\\\in B} x^b \\\\right| < '\n",
      " '(1-x)^{2011}.\\\\]')\n",
      "'\\\\text{Yes}'\n",
      "('The problem asks to determine whether two different sets A and B exist that '\n",
      " 'satisfy a given condition. Although the answer provided is a simple \"Yes,\" '\n",
      " 'the problem itself requires a proof to demonstrate the existence of such '\n",
      " 'sets.  Therefore, this is a Case 2 problem.\\n'\n",
      " '\\n'\n",
      " '[[2]]\\n')\n",
      "('Does there exist $ 2002$ distinct positive integers $ k_1, k_2, \\\\cdots '\n",
      " 'k_{2002}$ such that for any positive integer $ n \\\\geq 2001$, one of $ '\n",
      " 'k_12^n \\\\plus{} 1, k_22^n \\\\plus{} 1, \\\\cdots, k_{2002}2^n \\\\plus{} 1$ is '\n",
      " 'prime?')\n",
      "'\\\\text{No}'\n",
      "('The problem asks \"Does there exist\". This type of question is a yes/no '\n",
      " 'question, and hence it falls under Case 2. Although the answer provided is '\n",
      " 'simply \"No\", the problem itself requires a proof to show why no such '\n",
      " 'integers exist.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Does there exist a finite set $A$ of positive integers of at least two '\n",
      " 'elements and an infinite set $B$ of positive integers, such that any two '\n",
      " 'distinct elements in $A+B$ are coprime, and for any coprime positive '\n",
      " 'integers $m,n$, there exists an element $x$ in $A+B$ satisfying $x\\\\equiv n '\n",
      " '\\\\pmod m$ ?\\n'\n",
      " '\\n'\n",
      " 'Here $A+B=\\\\{a+b|a\\\\in A, b\\\\in B\\\\}$.')\n",
      "'\\\\text{No}'\n",
      "('The problem asks whether such sets exist, which is a yes/no question. This '\n",
      " 'makes it a proof-based problem, falling under Case 2. Although the answer is '\n",
      " 'a simple \"no\", the core of the problem lies in proving or disproving the '\n",
      " 'existence of the sets with the given properties.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('For any positive integer $d$, prove there are infinitely many positive '\n",
      " 'integers $n$ such that  $d(n!)-1$ is a composite number.')\n",
      "('\\\\text{There are infinitely many positive integers } n \\\\text{ such that } '\n",
      " 'd(n!) - 1 \\\\text{ is a composite number.}')\n",
      "('The problem asks to prove a statement. Specifically, it asks to prove that '\n",
      " 'there are infinitely many positive integers \\\\(n\\\\) such that \\\\(d(n!) - '\n",
      " '1\\\\) is a composite number for any positive integer \\\\(d\\\\).  The answer '\n",
      " 'implies that such integers exist. Thus, the problem falls under Case 2 '\n",
      " 'because it requires a proof.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: [[2]]\\n')\n",
      "('Let the circumcenter of triangle $ABC$ be $O$. $H_A$ is the projection of '\n",
      " '$A$ onto $BC$. The extension of $AO$ intersects the circumcircle of $BOC$ at '\n",
      " \"$A'$. The projections of $A'$ onto $AB, AC$ are $D,E$, and $O_A$ is the \"\n",
      " 'circumcentre of triangle $DH_AE$. Define $H_B, O_B, H_C, O_C$ similarly. \\n'\n",
      " 'Prove: $H_AO_A, H_BO_B, H_CO_C$ are concurrent')\n",
      "('\\\\text{The lines } H_AO_A, H_BO_B, \\\\text{ and } H_CO_C \\\\text{ are '\n",
      " 'concurrent at the orthocenter of } \\\\triangle H_AH_BH_C.')\n",
      "('The problem asks to prove a statement. The answer aims to prove this '\n",
      " 'statement. Thus, this is Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Let the intersections of $\\\\odot O_1$ and $\\\\odot O_2$ be $A$ and $B$. Point '\n",
      " '$R$ is on arc $AB$ of $\\\\odot O_1$ and $T$ is on arc $AB$ on $\\\\odot O_2$. '\n",
      " '$AR$ and $BR$ meet $\\\\odot O_2$ at $C$ and $D$; $AT$ and $BT$ meet $\\\\odot '\n",
      " 'O_1$ at $Q$ and $P$. If $PR$ and $TD$ meet at $E$ and $QR$ and $TC$ meet at '\n",
      " '$F$, then prove: $AE \\\\cdot BT \\\\cdot BR = BF \\\\cdot AT \\\\cdot AR$.')\n",
      "'AE \\\\cdot BT \\\\cdot BR = BF \\\\cdot AT \\\\cdot AR'\n",
      "('The problem asks to prove a specific geometric equation. This falls under '\n",
      " 'Case 2 because the objective is to demonstrate the validity of a given '\n",
      " 'statement through a proof. While the problem provides an equation, the core '\n",
      " 'task is not to find a solution or value but to establish the truth of the '\n",
      " 'relationship.  The \"answer\" simply restates what needs to be proven and '\n",
      " \"doesn't provide a numerical or descriptive solution.\\n\"\n",
      " '\\n'\n",
      " 'Final Answer: [[2]]\\n')\n",
      "('Given a fixed positive integer $a\\\\geq 9$. Prove: There exist finitely many '\n",
      " 'positive integers $n$, satisfying:\\n'\n",
      " '(1)$\\\\tau (n)=a$\\n'\n",
      " '(2)$n|\\\\phi (n)+\\\\sigma (n)$\\n'\n",
      " 'Note: For positive integer $n$, $\\\\tau (n)$ is the number of positive '\n",
      " 'divisors of $n$, $\\\\phi (n)$ is the number of positive integers $\\\\leq n$ '\n",
      " 'and relatively prime with $n$, $\\\\sigma (n)$ is the sum of positive divisors '\n",
      " 'of $n$.')\n",
      "'\\\\text{There exist finitely many positive integers } n.'\n",
      "('The problem asks to prove that there exist finitely many positive integers '\n",
      " '$n$ satisfying two given conditions. This clearly falls under Case 2, as it '\n",
      " 'requires a proof to show that a statement is true. Although the answer '\n",
      " 'states \"There exist finitely many positive integers $n$\", the core of the '\n",
      " 'problem lies in proving this statement, not finding a specific numerical '\n",
      " 'answer.\\n'(\n",
      " 'Determine whether or not there are any positive integral solutions of the '\n",
      " '\\n''simultaneous equations \\\\begin{align*} x_1^2 +x_2^2 +\\\\cdots +x_{1985}^2 & = '\n",
      " \n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n''y^3,\\\\\\\\ x_1^3 +x_2^3 +\\\\cdots +x_{1985}^3 & = z^2 \\\\end{align*} with ')\n",
      " \n",
      "'distinct integers $x_1,x_2,\\\\cdots,x_{1985}$ .')\n",
      "'A positive integral solution exists.'\n",
      "('The problem asks to determine whether or not positive integral solutions '\n",
      " 'exist for a given system of equations. While the problem does not explicitly '\n",
      " 'ask for the solutions themselves, the answer indicates that a solution '\n",
      " 'exists. This implies that the problem is primarily concerned with the '\n",
      " 'existence of a solution rather than a formal proof. Thus, it falls under '\n",
      " 'Case 1.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{1}$\\n')\n",
      "('Let $x_n=\\\\binom{2n}{n}$ for all $n\\\\in\\\\mathbb{Z}^+$. Prove there exist '\n",
      " 'infinitely many finite sets $A,B$ of positive integers, satisfying $A \\\\cap '\n",
      " 'B = \\\\emptyset $, and \\\\[\\\\frac{{\\\\prod\\\\limits_{i \\\\in A} {{x_i}} '\n",
      " '}}{{\\\\prod\\\\limits_{j\\\\in B}{{x_j}} }}=2012.\\\\]')\n",
      "'\\\\text{There exist infinitely many such sets } A \\\\text{ and } B.'\n",
      "('The problem asks to prove the existence of infinitely many sets $A$ and $B$ '\n",
      " \"with certain properties.  While a numerical calculation isn't the goal, the \"\n",
      " 'question is about proving a property (existence of infinitely many sets), '\n",
      " 'which makes this a proof-based problem. Thus, this is Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Attempt of a halfways nice solution.\\r\\n'\n",
      " '\\r\\n'\n",
      " '[color=blue][b]Problem.[/b] Let ABC be a triangle with $C\\\\geq 60^{\\\\circ}$. '\n",
      " 'Prove the inequality\\n'\n",
      " '\\n'\n",
      " '$\\\\left(a+b\\\\right)\\\\cdot\\\\left(\\\\frac{1}{a}+\\\\frac{1}{b}+\\\\frac{1}{c}\\\\right)\\\\geq '\n",
      " '4+\\\\frac{1}{\\\\sin\\\\frac{C}{2}}$.[/color]\\r\\n'\n",
      " '\\r\\n'\n",
      " '[i]Solution.[/i] First, we equivalently transform the inequality in '\n",
      " 'question:\\r\\n'\n",
      " '\\r\\n'\n",
      " '$\\\\left(a+b\\\\right)\\\\cdot\\\\left(\\\\frac{1}{a}+\\\\frac{1}{b}+\\\\frac{1}{c}\\\\right)\\\\geq '\n",
      " '4+\\\\frac{1}{\\\\sin\\\\frac{C}{2}}$\\r\\n'\n",
      " '$\\\\Longleftrightarrow\\\\ \\\\ \\\\ \\\\ \\\\ '\n",
      " '\\\\left(a+b\\\\right)\\\\cdot\\\\left(\\\\frac{1}{a}+\\\\frac{1}{b}\\\\right)+\\\\frac{a+b}{c}\\\\geq '\n",
      " '4+\\\\frac{1}{\\\\sin\\\\frac{C}{2}}$\\r\\n'\n",
      " '$\\\\Longleftrightarrow\\\\ \\\\ \\\\ \\\\ \\\\ '\n",
      " '\\\\left(a+b\\\\right)\\\\cdot\\\\left(\\\\frac{1}{a}+\\\\frac{1}{b}\\\\right)-4\\\\geq\\\\frac{1}{\\\\sin\\\\frac{C}{2}}-\\\\frac{a+b}{c}$\\r\\n'\n",
      " '$\\\\Longleftrightarrow\\\\ \\\\ \\\\ \\\\ \\\\ '\n",
      " '\\\\frac{\\\\left(a-b\\\\right)^2}{ab}\\\\geq\\\\frac{1}{\\\\sin\\\\frac{C}{2}}-\\\\frac{a+b}{c}$.\\r\\n'\n",
      " '\\r\\n'\n",
      " 'Now, by the Mollweide formulas,\\r\\n'\n",
      " '\\r\\n'\n",
      " '$\\\\frac{a+b}{c}=\\\\frac{\\\\cos\\\\frac{A-B}{2}}{\\\\sin\\\\frac{C}{2}}$ and '\n",
      " '$\\\\frac{a-b}{c}=\\\\frac{\\\\sin\\\\frac{A-B}{2}}{\\\\cos\\\\frac{C}{2}}$, so that\\r\\n'\n",
      " '$\\\\frac{a-b}{a+b}=\\\\frac{a-b}{c} : '\n",
      " '\\\\frac{a+b}{c}=\\\\frac{\\\\sin\\\\frac{A-B}{2}}{\\\\cos\\\\frac{C}{2}} : '\n",
      " '\\\\frac{\\\\cos\\\\frac{A-B}{2}}{\\\\sin\\\\frac{C}{2}}=\\\\frac{\\\\sin\\\\frac{A-B}{2}\\\\sin\\\\frac{C}{2}}{\\\\cos\\\\frac{A-B}{2}\\\\cos\\\\frac{C}{2}}$.\\r\\n'\n",
      " '\\r\\n'\n",
      " 'Now, $\\\\cos^2\\\\frac{C}{2}\\\\leq 1$ (as the square of every cosine is $\\\\leq '\n",
      " '1$). On the other hand, the AM-GM inequality yields '\n",
      " '$ab\\\\leq\\\\frac14\\\\left(a+b\\\\right)^2$. Hence,\\r\\n'\n",
      " '\\r\\n'\n",
      " '$\\\\frac{\\\\left(a-b\\\\right)^2}{ab}\\\\geq\\\\frac{\\\\left(a-b\\\\right)^2}{\\\\frac14\\\\left(a+b\\\\right)^2}$       '\n",
      " '(since $ab\\\\leq\\\\frac14\\\\left(a+b\\\\right)^2$)\\r\\n'\n",
      " '$=4\\\\left(\\\\frac{a-b}{a+b}\\\\right)^2=4\\\\left(\\\\frac{\\\\sin\\\\frac{A-B}{2}\\\\sin\\\\frac{C}{2}}{\\\\cos\\\\frac{A-B}{2}\\\\cos\\\\frac{C}{2}}\\\\right)^2=\\\\frac{4\\\\sin^2\\\\frac{A-B}{2}\\\\sin^2\\\\frac{C}{2}}{\\\\cos^2\\\\frac{A-B}{2}\\\\cos^2\\\\frac{C}{2}}$\\r\\n'\n",
      " '$\\\\geq\\\\frac{4\\\\sin^2\\\\frac{A-B}{2}\\\\sin^2\\\\frac{C}{2}}{\\\\cos^2\\\\frac{A-B}{2}}$         '\n",
      " '(since $\\\\cos^2\\\\frac{C}{2}\\\\leq 1$).\\r\\n'\n",
      " '$=\\\\frac{4\\\\left(2\\\\sin\\\\frac{A-B}{4}\\\\cos\\\\frac{A-B}{4}\\\\right)^2\\\\sin^2\\\\frac{C}{2}}{\\\\cos^2\\\\frac{A-B}{2}}=\\\\frac{16\\\\sin^2\\\\frac{A-B}{4}\\\\cos^2\\\\frac{A-B}{4}\\\\sin^2\\\\frac{C}{2}}{\\\\cos^2\\\\frac{A-B}{2}}$.\\r\\n'\n",
      " '\\r\\n'\n",
      " 'Thus, instead of proving the inequality '\n",
      " '$\\\\frac{\\\\left(a-b\\\\right)^2}{ab}\\\\geq\\\\frac{1}{\\\\sin\\\\frac{C}{2}}-\\\\frac{a+b}{c}$, '\n",
      " 'it will be enough to show the stronger inequality\\r\\n'\n",
      " '\\r\\n'\n",
      " '$\\\\frac{16\\\\sin^2\\\\frac{A-B}{4}\\\\cos^2\\\\frac{A-B}{4}\\\\sin^2\\\\frac{C}{2}}{\\\\cos^2\\\\frac{A-B}{2}}\\\\geq\\\\frac{1}{\\\\sin\\\\frac{C}{2}}-\\\\frac{a+b}{c}$.\\r\\n'\n",
      " '\\r\\n'\n",
      " 'Noting that\\r\\n'\n",
      " '\\r\\n'\n",
      " '$\\\\frac{1}{\\\\sin\\\\frac{C}{2}}-\\\\frac{a+b}{c}=\\\\frac{1}{\\\\sin\\\\frac{C}{2}}-\\\\frac{\\\\cos\\\\frac{A-B}{2}}{\\\\sin\\\\frac{C}{2}}=\\\\frac{1-\\\\cos\\\\frac{A-B}{2}}{\\\\sin\\\\frac{C}{2}}=\\\\frac{2\\\\sin^2\\\\frac{A-B}{4}}{\\\\sin\\\\frac{C}{2}}$,\\r\\n'\n",
      " '\\r\\n'\n",
      " 'we transform this inequality into\\r\\n'\n",
      " '\\r\\n'\n",
      " '$\\\\frac{16\\\\sin^2\\\\frac{A-B}{4}\\\\cos^2\\\\frac{A-B}{4}\\\\sin^2\\\\frac{C}{2}}{\\\\cos^2\\\\frac{A-B}{2}}\\\\geq\\\\frac{2\\\\sin^2\\\\frac{A-B}{4}}{\\\\sin\\\\frac{C}{2}}$,\\r\\n'\n",
      " '\\r\\n'\n",
      " 'what, upon multiplication by '\n",
      " '$\\\\frac{\\\\cos^2\\\\frac{A-B}{2}\\\\sin\\\\frac{C}{2}}{16\\\\sin^2\\\\frac{A-B}{4}}$ '\n",
      " 'and rearrangement of terms, becomes\\r\\n'\n",
      " '\\r\\n'\n",
      " '$\\\\sin^3\\\\frac{C}{2}\\\\cos^2\\\\frac{A-B}{4}\\\\geq\\\\frac18\\\\cos^2\\\\frac{A-B}{2}$.\\r\\n'\n",
      " '\\r\\n'\n",
      " 'But this trivially follows by multiplying the two inequalities\\r\\n'\n",
      " '\\r\\n'\n",
      " '$\\\\sin^3\\\\frac{C}{2}\\\\geq\\\\frac18$      (equivalent to '\n",
      " '$\\\\sin\\\\frac{C}{2}\\\\geq\\\\frac12$, what is true because $60^{\\\\circ}\\\\leq '\n",
      " 'C\\\\leq 180^{\\\\circ}$ yields $30^{\\\\circ}\\\\leq\\\\frac{C}{2}\\\\leq 90^{\\\\circ}$) '\n",
      " 'and\\r\\n'\n",
      " '$\\\\cos^2\\\\frac{A-B}{4}\\\\geq\\\\cos^2\\\\frac{A-B}{2}$    (follows from the '\n",
      " 'obvious fact that '\n",
      " '$\\\\left|\\\\frac{A-B}{4}\\\\right|\\\\leq\\\\left|\\\\frac{A-B}{2}\\\\right|$ since '\n",
      " '$\\\\left|\\\\frac{A-B}{2}\\\\right|<90^{\\\\circ}$, what is true because '\n",
      " '$\\\\left|A-B\\\\right|<180^{\\\\circ}$, as the angles A and B, being angles of a '\n",
      " 'triangle, lie between 0° and 180°).\\r\\n'\n",
      " '\\r\\n'\n",
      " 'Hence, the problem is solved.\\r\\n'\n",
      " '\\r\\n'\n",
      " '  Darij')\n",
      "('(a + b) \\\\left( \\\\frac{1}{a} + \\\\frac{1}{b} + \\\\frac{1}{c} \\\\right) \\\\geq 4 '\n",
      " '+ \\\\frac{1}{\\\\sin \\\\frac{C}{2}}')\n",
      "('The problem asks to prove an inequality given some conditions related to a '\n",
      " 'triangle.  While there is a provided solution (a proof), the problem itself '\n",
      " 'is about proving a specific inequality. This makes it fall under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{2}$\\n')\n",
      "('Does there exists a positive irrational number ${x},$ such that there are at '\n",
      " 'most finite positive integers ${n},$ satisfy that for any integer $1\\\\leq '\n",
      " 'k\\\\leq n,$ $\\\\{kx\\\\}\\\\geq\\\\frac 1{n+1}?$')\n",
      "'\\\\text{No}'\n",
      "('The problem asks whether a specific type of positive irrational number '\n",
      " 'exists.  The answer provided is \"No.\" This clearly falls under Case 2, as '\n",
      " 'the question is asking to prove/disprove a statement (in this case, disprove '\n",
      " 'the existence of such a number). While an answer is given, the core of the '\n",
      " 'problem lies in proving the non-existence, which requires a proof.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is [[2]]\\n')\n",
      "('In a fictional world, each resident (viewed as geometric point) is assigned '\n",
      " 'a number: $1,2, \\\\cdots$. In order to fight against some epidemic, the '\n",
      " 'residents take some vaccine and they stay at the vaccination site after '\n",
      " 'taking the shot for observation. Now suppose that the shape of the '\n",
      " 'Observation Room is a circle of radius $\\\\frac{1}{4}$, and one requires that '\n",
      " 'the distance $d_{m, n}$ between the Resident No. $m$ and the Resident No. '\n",
      " '$n$ must satisfy $(m+n) d_{m, n} \\\\geq 1$. Where we consider the distance on '\n",
      " 'the circle, i.e., the length of the minor arc between two points. Proof '\n",
      " 'Question: Give a proof of your answer to Question (i).')\n",
      "'The circle can accommodate any quantity of residents.'\n",
      "('The problem asks for a proof that any number of residents can fit in the '\n",
      " 'observation room given the constraint $(m+n)d_{m,n} \\\\ge 1$. This is a proof '\n",
      " 'question with a yes/no answer (yes, any number can fit). Thus, this falls '\n",
      " 'under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: [[2]]\\n')\n",
      "('A convex hexagon $A B C D E F$ is inscribed in a circle. Prove the '\n",
      " 'inequality $A C \\\\cdot B D \\\\cdot C E \\\\cdot D F \\\\cdot A E \\\\cdot B F \\\\geq '\n",
      " '27 A B \\\\cdot B C \\\\cdot C D \\\\cdot D E \\\\cdot E F \\\\cdot F A$.')\n",
      "('\\\\[\\n'\n",
      " 'A C \\\\cdot B D \\\\cdot C E \\\\cdot D F \\\\cdot A E \\\\cdot B F \\\\geq 27 A B '\n",
      " '\\\\cdot B C \\\\cdot C D \\\\cdot D E \\\\cdot E F \\\\cdot F A\\n'\n",
      " '\\\\]')\n",
      "('The problem asks to prove an inequality. This clearly falls under Case 2, '\n",
      " 'problems that require a proof to answer Yes/No or prove/disprove a '\n",
      " 'statement.  Specifically, it asks to prove the given inequality holds for a '\n",
      " 'convex hexagon inscribed in a circle.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('The area of a convex pentagon $A B C D E$ is $S$, and the circumradii of the '\n",
      " 'triangles $A B C, B C D, C D E, D E A, E A B$ are $R_{1}, R_{2}, R_{3}, '\n",
      " 'R_{4}, R_{5}$. Prove the inequality '\n",
      " '$R_{1}^{4}+R_{2}^{4}+R_{3}^{4}+R_{4}^{4}+R_{5}^{4} \\\\geqslant \\\\frac{4}{5 '\n",
      " '\\\\sin ^{2} 108^{\\\\circ}} S^{2}$.')\n",
      "('\\\\[\\n'\n",
      " 'R_{1}^{4} + R_{2}^{4} + R_{3}^{4} + R_{4}^{4} + R_{5}^{4} \\\\geq \\\\frac{4}{5 '\n",
      " '\\\\sin^{2} 108^{\\\\circ}} S^{2}\\n'\n",
      " '\\\\]')\n",
      "('The problem asks to prove an inequality relating the circumradii of '\n",
      " 'triangles formed by vertices of a convex pentagon to the area of the '\n",
      " 'pentagon. This clearly falls under Case 2, as the objective is to prove a '\n",
      " 'specific relationship rather than finding a specific value or set of '\n",
      " 'values.  The presence of the word \"prove\" reinforces this classification.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: [[2]]\\n')\n",
      "('FIx positive integer $n$. Prove: For any positive integers $a,b,c$ not '\n",
      " 'exceeding $3n^2+4n$, there exist integers $x,y,z$ with absolute value not '\n",
      " 'exceeding $2n$ and not all $0$, such that $ax+by+cz=0$')\n",
      "'0'\n",
      "('The problem asks to prove a statement. Thus, this is Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is [[2]]\\n')\n",
      "('Does there exist a two-variable polynomial $P(x, y)$ with real number '\n",
      " 'coefficients such that $P(x, y)$ is positive exactly when $x$ and $y$ are '\n",
      " 'both positive?')\n",
      "'No such polynomial exists.'\n",
      "('The problem asks whether a polynomial exists with a specific property. '\n",
      " 'Although the answer provided includes a proof to show that no such '\n",
      " 'polynomial exists, the problem itself is a yes/no question. Therefore, this '\n",
      " 'falls under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Does there exist a field such that its multiplicative group is isomorphic to '\n",
      " 'its additive group?')\n",
      "'There exist no such field.'\n",
      "('The problem asks whether a field exists with its multiplicative and additive '\n",
      " 'groups being isomorphic. While the answer provided states \"no,\" the core '\n",
      " 'question is about the existence of such a field, which makes this a '\n",
      " 'proof-based problem. Therefore, this falls under Case 2.\\n'\n",
      " '\\n'\n",
      " '[[2]]\\n')\n",
      "('Does there exist a real $3 \\\\times 3$ matrix $A$ such that '\n",
      " '\\\\operatorname{tr}(\\\\mathrm{A})=0$ and $A^{2}+A^{t}=I$ ? (tr $(\\\\mathrm{A})$ '\n",
      " 'denotes the trace of $A$, $A^{t}$ is the transpose of $A$, and $I$ is the '\n",
      " 'identity matrix.)')\n",
      "('There does not exist a real $3 \\\\times 3$ matrix $A$ such that '\n",
      " '$\\\\operatorname{tr}(A) = 0$ and $A^2 + A^t = I$.')\n",
      "('The problem asks whether a matrix with certain properties exists.  Although '\n",
      " 'the answer provides a proof to arrive at the answer \"no\", the problem itself '\n",
      " 'is simply asking whether such a matrix exists. This makes it a Case 2 '\n",
      " 'problem.\\n'\n",
      " '\\n'\n",
      " '[[2]]\\n')\n",
      "('There are $n$ line segments on the plane, no three intersecting at a point, '\n",
      " 'and each pair intersecting once in their respective interiors. Tony and his '\n",
      " '$2 n-1$ friends each stand at a distinct endpoint of a line segment. Tony '\n",
      " 'wishes to send Christmas presents to each of his friends as follows: First, '\n",
      " 'he chooses an endpoint of each segment as a \"sink\". Then he places the '\n",
      " 'present at the endpoint of the segment he is at. The present moves as '\n",
      " 'follows: - If it is on a line segment, it moves towards the sink. - When it '\n",
      " 'reaches an intersection of two segments, it changes the line segment it '\n",
      " 'travels on and starts moving towards the new sink. If the present reaches an '\n",
      " 'endpoint, the friend on that endpoint can receive their present. Prove Tony '\n",
      " 'can send presents to exactly $n$ of his $2 n-1$ friends.')\n",
      "'Tony can send presents to exactly \\\\( n \\\\) of his \\\\( 2n-1 \\\\) friends.'\n",
      "('The problem asks to prove that Tony can send presents to exactly \\\\(n\\\\) of '\n",
      " 'his \\\\(2n-1\\\\) friends. This clearly falls under Case 2, as it requires a '\n",
      " 'proof to show a statement. The answer provided also explicitly gives a proof '\n",
      " 'for the problem.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{2}$\\n')\n",
      "('Determine whether there exists a positive integer \\\\(n\\\\) for which '\n",
      " '\\\\(g(n)>n^{0.999 n}\\\\), where \\\\(f(n), g(n)\\\\) are the minimal positive '\n",
      " 'integers such that '\n",
      " '\\\\(1+\\\\frac{1}{1!}+\\\\frac{1}{2!}+\\\\ldots+\\\\frac{1}{n!}=\\\\frac{f(n)}{g(n)}\\\\).')\n",
      "'There exists a positive integer \\\\( n \\\\) for which \\\\( g(n) > n^{0.999n} \\\\).'\n",
      "('The problem asks to determine whether a positive integer $n$ exists '\n",
      " 'satisfying a given condition. While the answer states that such an integer '\n",
      " 'exists, the problem itself is a yes/no question. Thus, it belongs to Case '\n",
      " '2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('A regular tetrahedron $A B C D$ and points $M, N$ are given in space. Prove '\n",
      " 'the inequality $M A \\\\cdot N A+M B \\\\cdot N B+M C \\\\cdot N C \\\\geqslant M D '\n",
      " '\\\\cdot N D$')\n",
      "'\\\\[ MA \\\\cdot NA + MB \\\\cdot NB + MC \\\\cdot NC \\\\geq MD \\\\cdot ND \\\\]'\n",
      "('The problem asks to prove an inequality. This clearly falls under Case 2, '\n",
      " 'which involves proving or disproving a statement.  The presence of an answer '\n",
      " \"doesn't change the category.\\n\"\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Let $p$ be a prime number. Prove the following theorem of Euler: the '\n",
      " 'equation $p=x^{2}+3 y^{2}$ has a solution with $x, y \\\\in \\\\mathbb{Z}$ if '\n",
      " 'and only if $p=3$ or $p \\\\equiv 1(\\\\bmod 3)$. (You may use the fact that the '\n",
      " 'ring of integers of $\\\\mathbb{Q}(\\\\sqrt{-3})$ is a principal ideal domain.)')\n",
      "('The equation \\\\( p = x^2 + 3y^2 \\\\) has a solution with \\\\( x, y \\\\in '\n",
      " '\\\\mathbb{Z} \\\\) if and only if \\\\( p = 3 \\\\) or \\\\( p \\\\equiv 1 \\\\pmod{3} '\n",
      " '\\\\).')\n",
      "('The problem asks to prove a statement of the form \"A if and only if B\". This '\n",
      " 'matches the criteria for Case 2. While the answer reaffirms the statement '\n",
      " 'that needs to be proven, it does not provide a direct solution. The problem '\n",
      " 'explicitly requests a proof, thus making it fall under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is [[2]]\\n')\n",
      "('(a) Does there exist a finite set of points, not all collinear, such that a '\n",
      " 'line between any two points in the set passes through a third point in the '\n",
      " 'set? (b) Let $ABC$ be a triangle and $P$ be a point. The isogonal conjugate '\n",
      " 'of $P$ is the intersection of the reflection of line $AP$ over the $A$-angle '\n",
      " 'bisector, the reflection of line $BP$ over the $B$-angle bisector, and the '\n",
      " 'reflection of line $CP$ over the $C$-angle bisector. Clearly the incenter is '\n",
      " 'its own isogonal conjugate. Does there exist another point that is its own '\n",
      " 'isogonal conjugate? (c) Let $F$ be a convex figure in a plane, and let $P$ '\n",
      " 'be the largest pentagon that can be inscribed in $F$. Is it necessarily true '\n",
      " 'that the area of $P$ is at least $\\\\frac{3}{4}$ the area of $F$? (d) Is it '\n",
      " 'possible to cut an equilateral triangle into 2017 pieces, and rearrange the '\n",
      " 'pieces into a square? (e) Let $ABC$ be an acute triangle and $P$ be a point '\n",
      " 'in its interior. Let $D, E, F$ lie on $BC, CA, AB$ respectively so that $PD$ '\n",
      " 'bisects $\\\\angle BPC, PE$ bisects $\\\\angle CPA$, and $PF$ bisects $\\\\angle '\n",
      " 'APB$. Is it necessarily true that $AP+BP+CP \\\\geq 2(PD+PE+PF)$? (f) Let '\n",
      " '$P_{2018}$ be the surface area of the 2018-dimensional unit sphere, and let '\n",
      " '$P_{2017}$ be the surface area of the 2017-dimensional unit sphere. Is '\n",
      " '$P_{2018}>P_{2017}$?')\n",
      "'NYYYYN'\n",
      "('The problem presents a series of questions, each of which can be answered '\n",
      " 'with \"yes\" or \"no\".  Although some of these questions might involve proofs '\n",
      " 'to justify the yes/no answer, the ultimate goal is simply to determine '\n",
      " 'whether the statement is true or false. Thus, these are all yes/no '\n",
      " 'questions. This falls under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: [[2]]\\n')\n",
      "('Does there exist an irrational number $\\\\alpha>1$ such that '\n",
      " '\\\\(\\\\left\\\\lfloor\\\\alpha^{n}\\\\right\\\\rfloor \\\\equiv 0 \\\\quad(\\\\bmod 2017)\\\\) '\n",
      " 'for all integers $n \\\\geq 1$ ?')\n",
      "'Yes'\n",
      "('The problem asks whether an irrational number $\\\\alpha > 1$ exists such that '\n",
      " '$\\\\lfloor \\\\alpha^n \\\\rfloor \\\\equiv 0 \\\\pmod{2017}$ for all integers $n '\n",
      " '\\\\geq 1$.  While the answer provided is a simple \"yes,\" the problem itself '\n",
      " 'is asking to prove the existence of such a number.  Thus, this falls under '\n",
      " 'Case 2.\\n'\n",
      " '\\n'\n",
      " '[[2]]\\n')\n",
      "('(a) Can 1000 queens be placed on a $2017 \\\\times 2017$ chessboard such that '\n",
      " 'every square is attacked by some queen? A square is attacked by a queen if '\n",
      " 'it lies in the same row, column, or diagonal as the queen. (b) A $2017 '\n",
      " '\\\\times 2017$ grid of squares originally contains a 0 in each square. At any '\n",
      " 'step, Kelvin the Frog chooses two adjacent squares (two squares are adjacent '\n",
      " 'if they share a side) and increments the numbers in both of them by 1. Can '\n",
      " 'Kelvin make every square contain a different power of 2? (c) A tournament '\n",
      " 'consists of single games between every pair of players, where each game has '\n",
      " 'a winner and loser with no ties. A set of people is dominated if there '\n",
      " 'exists a player who beats all of them. Does there exist a tournament in '\n",
      " 'which every set of 2017 people is dominated? (d) Every cell of a $19 \\\\times '\n",
      " '19$ grid is colored either red, yellow, green, or blue. Does there '\n",
      " 'necessarily exist a rectangle whose sides are parallel to the grid, all of '\n",
      " 'whose vertices are the same color? (e) Does there exist a $c \\\\in '\n",
      " '\\\\mathbb{R}^{+}$such that $\\\\max (|A \\\\cdot A|,|A+A|) \\\\geq c|A| \\\\log '\n",
      " '^{2}|A|$ for all finite sets $A \\\\subset \\\\mathbb{Z}$? (f) Can the set '\n",
      " '$\\\\{1,2, \\\\ldots, 1093\\\\}$ be partitioned into 7 subsets such that each '\n",
      " 'subset is sum-free (i.e. no subset contains $a, b, c$ with $a+b=c)$?')\n",
      "'NNYYYY'\n",
      "('The problem presents a series of questions, each of which can be answered '\n",
      " 'with \"yes\" or \"no.\"  Although some may require extensive work to determine '\n",
      " 'the correct answer, each question has a definitive \"yes\" or \"no\" response.  '\n",
      " 'This puts the problem squarely in Case 2, as each question asks to prove or '\n",
      " 'disprove a statement.  The provided answer confirms this by giving a '\n",
      " 'sequence of \"yes\" and \"no\" answers.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: [[2]]\\n')\n",
      "('Let $S=\\\\{2,3,4, \\\\ldots\\\\}$ denote the set of integers that are greater '\n",
      " 'than or equal to 2. Does there exist a function $f: S \\\\rightarrow S$ such '\n",
      " 'that $f(a) f(b)=f\\\\left(a^{2} b^{2}\\\\right)$ for all $a, b \\\\in S$ with $a '\n",
      " '\\\\neq b$?')\n",
      "'There is no such function \\\\( f: S \\\\rightarrow S \\\\).'\n",
      "('The problem asks whether a function with a specified property exists.  While '\n",
      " 'the answer provides a proof to show that such a function does *not* exist, '\n",
      " 'the question itself is a yes/no question about existence. Thus, this falls '\n",
      " 'under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Let $k \\\\geq 14$ be an integer, and let $p_{k}$ be the largest prime number '\n",
      " 'which is strictly less than $k$. You may assume that $p_{k} \\\\geq 3 k / 4$. '\n",
      " 'Let $n$ be a composite integer. Prove: (a) if $n=2 p_{k}$, then $n$ does not '\n",
      " 'divide $(n-k)$ !; (b) if $n>2 p_{k}$, then $n$ divides $(n-k)$ ! .')\n",
      "'(a) $2 p_{k} \\\\nmid (n-k)!$\\n\\n(b) $n \\\\mid (n-k)!$'\n",
      "('The problem asks to *prove* two statements. Thus, this is a Case 2 problem.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Does there exist a continuously differentiable function $f: \\\\mathbb{R} '\n",
      " '\\\\rightarrow \\\\mathbb{R}$ such that for every $x \\\\in \\\\mathbb{R}$ we have '\n",
      " '$f(x)>0$ and $f^{\\\\prime}(x)=f(f(x))$ ?')\n",
      "('There does not exist a continuously differentiable function \\\\( f: '\n",
      " '\\\\mathbb{R} \\\\rightarrow \\\\mathbb{R} \\\\) such that for every \\\\( x \\\\in '\n",
      " '\\\\mathbb{R} \\\\) we have \\\\( f(x) > 0 \\\\) and \\\\( f^{\\\\prime}(x) = f(f(x)) '\n",
      " '\\\\).')\n",
      "('The problem asks whether a function with specific properties exists. While '\n",
      " 'the answer provides a proof to justify the answer \"no\", the core of the '\n",
      " 'problem is a yes/no question about existence.  Thus, this is a Case 2 '\n",
      " 'problem.\\n'\n",
      " '\\n'\n",
      " '[[2]]\\n')\n",
      "('Determine whether or not there exist 15 integers $m_{1}, \\\\ldots, m_{15}$ '\n",
      " 'such that $\\\\sum_{k=1}^{15} m_{k} \\\\cdot \\\\arctan (k)=\\\\arctan (16)$.')\n",
      "('There do not exist 15 integers \\\\( m_{1}, \\\\ldots, m_{15} \\\\) such that \\\\( '\n",
      " '\\\\sum_{k=1}^{15} m_{k} \\\\cdot \\\\arctan (k) = \\\\arctan (16) \\\\).')\n",
      "('The problem asks to determine whether or not there exist 15 integers that '\n",
      " 'satisfy a given equation.  The answer is a definitive \"no,\" Thus, this is a '\n",
      " 'Case 2 problem as it requires a proof to answer Yes/No.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{2}$\\n')\n",
      "('(a) Does $\\\\sum_{i=1}^{p-1} \\\\frac{1}{i} \\\\equiv 0\\\\left(\\\\bmod '\n",
      " 'p^{2}\\\\right)$ for all odd prime numbers $p$? (Note that $\\\\frac{1}{i}$ '\n",
      " 'denotes the number such that $\\\\left.i \\\\cdot \\\\frac{1}{i} \\\\equiv '\n",
      " '1\\\\left(\\\\bmod p^{2}\\\\right)\\\\right)$ (b) Do there exist 2017 positive '\n",
      " 'perfect cubes that sum to a perfect cube? (c) Does there exist a right '\n",
      " 'triangle with rational side lengths and area 5? (d) A magic square is a $3 '\n",
      " '\\\\times 3$ grid of numbers, all of whose rows, columns, and major diagonals '\n",
      " 'sum to the same value. Does there exist a magic square whose entries are all '\n",
      " 'prime numbers? (e) Is $\\\\prod_{p} '\n",
      " '\\\\frac{p^{2}+1}{p^{2}-1}=\\\\frac{2^{2}+1}{2^{2}-1} \\\\cdot '\n",
      " '\\\\frac{3^{2}+1}{3^{2}-1} \\\\cdot \\\\frac{5^{2}+1}{5^{2}-1} \\\\cdot '\n",
      " '\\\\frac{7^{2}+1}{7^{2}-1} \\\\cdot \\\\ldots$ a rational number? (f) Do there '\n",
      " 'exist an infinite number of pairs of distinct integers $(a, b)$ such that '\n",
      " '$a$ and $b$ have the same set of prime divisors, and $a+1$ and $b+1$ also '\n",
      " 'have the same set of prime divisors?')\n",
      "'NYYYYY'\n",
      "('The problem presents a series of questions, each of which can be answered '\n",
      " 'with \"Yes\" or \"No\".  Although some of these questions might involve complex '\n",
      " 'mathematics to determine the answer, each ultimately has a definitive \"Yes\" '\n",
      " 'or \"No\" answer provided, placing it firmly in Case 2.  The provided answer '\n",
      " 'confirms this by providing a sequence of Y/N answers.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Let $X_{1}, \\\\cdots, X_{n}$ be $n$ independent and identically distributed '\n",
      " 'observations from the Cauchy distribution with density function '\n",
      " '$f(x)=\\\\frac{1}{\\\\pi} \\\\frac{1}{1+(x-\\\\theta)^{2}}, x \\\\in \\\\mathbb{R}$. b) '\n",
      " 'Can you find an unbiased estimator $T$ that attains the lower bound in part '\n",
      " 'a)? If yes, please construct one. If no, please show why such an estimator '\n",
      " 'does not exist.')\n",
      "'No unbiased estimator exists that attains the lower bound.'\n",
      "('The problem asks whether an unbiased estimator exists and to justify the '\n",
      " 'answer.  This is a yes/no question that requires a proof/justification, '\n",
      " 'which falls under Case 2.\\n'\n",
      " '\\n'\n",
      " '[[2]]\\n')\n",
      "('We know that $2021=43 \\\\times 47$. Is there a polyhedron whose surface can '\n",
      " 'be formed by gluing together 43 equal non-planar 47-gons?')\n",
      "'YES'\n",
      "('The problem asks a yes/no question. Although the answer is YES, the problem '\n",
      " 'itself is a proof-based problem. Thus, this falls under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Prove or disprove the following statements: (a) There exists a monotone '\n",
      " 'function $f:[0,1] \\\\rightarrow[0,1]$ such that for each $y \\\\in[0,1]$ the '\n",
      " 'equation $f(x)=y$ has uncountably many solutions $x$. (b) There exists a '\n",
      " 'continuously differentiable function $f:[0,1] \\\\rightarrow[0,1]$ such that '\n",
      " 'for each $y \\\\in[0,1]$ the equation $f(x)=y$ has uncountably many solutions '\n",
      " '$x$.')\n",
      "'a. False, b. False'\n",
      "('The problem presents two statements and asks to prove or disprove them. This '\n",
      " 'clearly falls under Case 2, as it requires proofs to determine whether the '\n",
      " 'statements are true or false. Although the answer provides the final result '\n",
      " '(False for both), the core of the problem lies in constructing the proofs '\n",
      " 'themselves.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('For any positive integer $m$, denote by $P(m)$ the product of positive '\n",
      " 'divisors of $m$ (e.g. $P(6)=36$ ). For every positive integer $n$ define the '\n",
      " 'sequence $$a_{1}(n)=n, \\\\quad a_{k+1}(n)=P\\\\left(a_{k}(n)\\\\right) '\n",
      " '\\\\quad(k=1,2, \\\\ldots, 2016) .$$ Determine whether for every set $S '\n",
      " '\\\\subseteq\\\\{1,2, \\\\ldots, 2017\\\\}$, there exists a positive integer $n$ '\n",
      " 'such that the following condition is satisfied: For every $k$ with $1 \\\\leq '\n",
      " 'k \\\\leq 2017$, the number $a_{k}(n)$ is a perfect square if and only if $k '\n",
      " '\\\\in S$.')\n",
      "('Yes, such a positive integer $n$ exists for every set $S \\\\subseteq \\\\{1,2, '\n",
      " '\\\\ldots, 2017\\\\}$.')\n",
      "('The problem asks to determine whether for every subset \\\\(S\\\\) of \\\\(\\\\{1, '\n",
      " '2, \\\\ldots, 2017\\\\}\\\\), there exists a positive integer \\\\(n\\\\) such that '\n",
      " '\\\\(a_k(n)\\\\) is a perfect square if and only if \\\\(k \\\\in S\\\\).  The answer '\n",
      " 'provided is \"yes\". This means the problem is asking to prove the existence '\n",
      " 'of such an integer \\\\(n\\\\) for every subset \\\\(S\\\\). Thus, it falls under '\n",
      " 'Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is [[2]]\\n')\n",
      "('Does there exist an infinite set $M$ consisting of positive integers such '\n",
      " 'that for any $a, b \\\\in M$, with $a<b$, the sum $a+b$ is square-free? (A '\n",
      " 'positive integer is called square-free if no perfect square greater than 1 '\n",
      " 'divides it.)')\n",
      "'Yes, such a set exists.'\n",
      "('The problem asks whether an infinite set M with a specific property exists. '\n",
      " 'While the answer provided is a \"yes,\" the core of the problem is about '\n",
      " 'proving the existence of such a set. This makes it fall under Case 2.\\n'\n",
      " '\\n'\n",
      " '[[2]]\\n')\n",
      "('A mass of 15 kg is halfway between 10 kg and 20 kg on the vertical axis. '\n",
      " 'What is the age of the cod when its mass is 15 kg?')\n",
      "'7'\n",
      "('The problem presents information about masses and their positions on a '\n",
      " 'vertical axis, but then abruptly asks for the age of a cod when its mass is '\n",
      " '15 kg. There is no apparent connection between the initial information about '\n",
      " \"masses and the age of a cod. This seems like a non sequitur and doesn't \"\n",
      " 'constitute a coherent math problem.  It might be a trick question or a '\n",
      " 'riddle. The provided answer \"7\" doesn\\'t have any clear derivation from the '\n",
      " 'problem statement.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: [[3]]\\n')\n",
      "('Is there a strictly increasing function $f: \\\\mathbb{R} \\\\to \\\\mathbb{R}$ '\n",
      " \"such that $f'(x) = f(f(x))$ for all $x$?\")\n",
      "'No'\n",
      "('The problem asks whether a function with a specific property exists. '\n",
      " 'Although the answer is a simple \"no,\" the problem requires a proof to '\n",
      " 'justify this answer.  Therefore, this is a Case 2 problem.\\n'\n",
      " '\\n'\n",
      " '[[2]]\\n')\n",
      "('Is there an infinite sequence of real numbers $a_1, a_2, a_3, \\\\dots$ such '\n",
      " 'that\\n'\n",
      " '\\\\[\\n'\n",
      " 'a_1^m + a_2^m + a_3^m + \\\\cdots = m\\n'\n",
      " '\\\\]\\n'\n",
      " 'for every positive integer $m$?')\n",
      "'No'\n",
      "('The problem asks whether such a sequence exists. This is a yes/no question, '\n",
      " 'and the answer provided is \"No.\" Thus, this problem falls under Case 2, '\n",
      " 'which includes problems that require a proof to answer Yes/No.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Is there a finite abelian group $G$ such that the product of the\\n'\n",
      " 'orders of all its elements is $2^{2009}$?')\n",
      "'No'\n",
      "('The problem asks a yes/no question, and the answer provided is \"No\". This '\n",
      " 'strongly suggests that the problem requires a proof to justify the answer.  '\n",
      " 'Therefore, this is a Case 2 problem.\\n'\n",
      " '\\n'\n",
      " '[[2]]\\n')\n",
      "('Let $h(x,y)$ be a real-valued function that is twice continuously '\n",
      " 'differentiable throughout $\\\\mathbb{R}^2$, and define\\n'\n",
      " '\\\\[\\\\rho(x,y) = yh_x - xh_y.\\\\]\\n'\n",
      " 'Prove or disprove: For any positive constants $d$ and $r$ with $d>r$, there '\n",
      " 'is a circle $\\\\mathcal{S}$ of radius $r$ whose center is a distance $d$ away '\n",
      " 'from the origin such that the integral of $\\\\rho$ over the interior of '\n",
      " '$\\\\mathcal{S}$ is zero.')\n",
      "'Proven: such a circle \\\\mathcal{S} exists.'\n",
      "('The problem asks to prove or disprove the existence of a circle with certain '\n",
      " 'properties such that the integral of a given function over the interior of '\n",
      " 'the circle is zero. Although the problem statement includes \"prove or '\n",
      " 'disprove,\" the answer indicates that a proof exists, confirming the '\n",
      " 'existence of such a circle. Therefore, the problem falls under Case 2, as it '\n",
      " 'requires a proof to answer yes/no to the existence question.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: [[2]]\\n')\n",
      "('Whether there are integers $a_1$, $a_2$, $\\\\cdots$, that are different from '\n",
      " 'each other, satisfying:\\n'\n",
      " '(1) For $\\\\forall k\\\\in\\\\mathbb N_+$, $a_{k^2}>0$ and $a_{k^2+k}<0$;\\n'\n",
      " '(2) For $\\\\forall n\\\\in\\\\mathbb N_+$, $\\\\left| a_{n+1}-a_n\\\\right|\\\\leqslant '\n",
      " '2023\\\\sqrt n$?')\n",
      "'\\\\text{No}'\n",
      "('The problem asks whether there exist integers $a_1, a_2, \\\\dots$ satisfying '\n",
      " 'two conditions. Although the answer provided is simply \"No\", the problem '\n",
      " 'itself is asking to prove/disprove the existence of such integers. Thus, it '\n",
      " 'falls under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Let $f : \\\\mathbb Q \\\\to \\\\mathbb Q$ be a function such that for any $x,y '\n",
      " '\\\\in \\\\mathbb Q$, the number $f(x+y)-f(x)-f(y)$ is an integer. Decide '\n",
      " 'whether it follows that there exists a constant $c$ such that $f(x) - cx$ is '\n",
      " 'an integer for every rational number $x$.')\n",
      "'\\\\text{No}'\n",
      "('The problem asks to decide whether a statement is true or false (\"Decide '\n",
      " 'whether it follows that...\"). This clearly falls under Case 2, where the '\n",
      " 'problem presents a statement, and we need to prove or disprove it. Although '\n",
      " 'the provided answer is simply \"No\", the core task is to provide a proof for '\n",
      " \"why the statement doesn't hold.\\n\"\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('( Gregory Galperin ) A square grid on the Euclidean plane consists of all '\n",
      " 'points  $(m,n)$ , where $m$ and $n$ are integers .  Is it possible to cover '\n",
      " 'all grid points by an infinite family of discs with non-overlapping '\n",
      " 'interiors if each disc in the family has radius at least 5?')\n",
      "('It is not possible to cover all grid points by an infinite family of discs '\n",
      " 'with non-overlapping interiors if each disc in the family has radius at '\n",
      " 'least 5.')\n",
      "('The problem asks whether it is possible to cover all grid points with discs '\n",
      " 'of radius at least 5. The answer is a direct \"no\". This falls under Case 2, '\n",
      " 'as the problem statement is equivalent to asking to prove/disprove the '\n",
      " 'statement.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: [[2]]\\n')\n",
      "('$(*)$ Let $ABC$ be a triangle with $\\\\angle ABC$ obtuse. The $A$ -excircle '\n",
      " 'is a circle in the exterior of $\\\\triangle ABC$ that is tangent to side '\n",
      " '$\\\\overline{BC}$ of the triangle and tangent to the extensions of the other '\n",
      " 'two sides. Let $E$ , $F$ be the feet of the altitudes from $B$ and $C$ to '\n",
      " 'lines $AC$ and $AB$ , respectively. Can line $EF$ be tangent to the $A$ '\n",
      " '-excircle?')\n",
      "'No, line $EF$ cannot be tangent to the $A$-excircle.'\n",
      "('The problem asks \"Can line EF be tangent to the A-excircle?\". This is a '\n",
      " 'yes/no question and the answer provided attempts to prove that the answer is '\n",
      " 'no. Thus, this is Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Do there exist 16 three digit numbers, using only three different digits in '\n",
      " 'all, so that the all numbers give different residues when divided by 16?')\n",
      "'It is impossible to select 16 such numbers.'\n",
      "('The problem asks whether there exist 16 three-digit numbers satisfying a '\n",
      " 'specific condition. The answer is a definitive \"No.\" Thus, this problem '\n",
      " 'falls under Case 2, requiring a proof to demonstrate the impossibility.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: [[2]]\\n')\n",
      "('Are there integers $a$ and $b$ such that $a^5b+3$ and $ab^5+3$ are both '\n",
      " 'perfect cubes of integers?')\n",
      "'No, such integers do not exist.'\n",
      "('The problem asks whether there exist integers $a$ and $b$ satisfying a given '\n",
      " 'property. This is a yes/no question, and the answer provided includes a '\n",
      " 'proof showing that the answer is no. Therefore, this is a Case 2 problem.\\n'\n",
      " '\\n'\n",
      " '[[2]]\\n')\n",
      "('Is the set of positive integers $n$ such that $n!+1$ divides (2012n)! finite '\n",
      " 'or infinite?')\n",
      "('The set of positive integers \\\\( n \\\\) such that \\\\( n! + 1 \\\\) divides \\\\( '\n",
      " '(2012n)! \\\\) is finite.')\n",
      "('The problem asks to determine whether the set of positive integers '\n",
      " 'satisfying a given divisibility condition is finite or infinite. This is a '\n",
      " 'yes/no question, which suggests Case 2. While the answer states that the set '\n",
      " 'is finite, the problem itself requires a proof to justify this claim. '\n",
      " 'Therefore, this falls under Case 2.\\n'\n",
      " '\\n'\n",
      " '[[2]]\\n')\n",
      "('Is it possible for the projection of the set of points $(x, y, z)$ with $0 '\n",
      " '\\\\leq x, y, z \\\\leq 1$ onto some two-dimensional plane to be a simple convex '\n",
      " 'pentagon?')\n",
      "'It is not possible.'\n",
      "('The problem asks whether it is possible for the projection of a set of '\n",
      " 'points onto a plane to be a simple convex pentagon. The answer provided is a '\n",
      " 'simple \"no.\" This indicates that the problem is asking for a proof to a '\n",
      " 'yes/no question. This falls under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Chim Tu has a large rectangular table. On it, there are finitely many pieces '\n",
      " 'of paper with nonoverlapping interiors, each one in the shape of a convex '\n",
      " 'polygon. At each step, Chim Tu is allowed to slide one piece of paper in a '\n",
      " 'straight line such that its interior does not touch any other piece of paper '\n",
      " 'during the slide. Can Chim Tu always slide all the pieces of paper off the '\n",
      " 'table in finitely many steps?')\n",
      "('Yes, Chim Tu can always slide all the pieces of paper off the table in '\n",
      " 'finitely many steps.')\n",
      "(\"The problem asks whether it's always possible to slide all pieces of paper \"\n",
      " 'off the table, and the answer provided is a simple \"yes.\" This indicates '\n",
      " 'that the problem is asking for a proof demonstrating the existence of a '\n",
      " \"strategy or algorithm to achieve this goal.  The problem itself doesn't have \"\n",
      " 'a direct numerical answer or a function to derive, but rather seeks a '\n",
      " 'justification for why such a process is always achievable. Therefore, this '\n",
      " 'falls under Case 2.\\n'\n",
      " '\\n'\n",
      " '[[2]]\\n')\n",
      "('Let $\\\\mathbb{N}=\\\\{1,2,3, \\\\ldots\\\\}$ be the set of all positive integers, '\n",
      " 'and let $f$ be a bijection from $\\\\mathbb{N}$ to $\\\\mathbb{N}$. Must there '\n",
      " 'exist some positive integer $n$ such that $(f(1), f(2), \\\\ldots, f(n))$ is a '\n",
      " 'permutation of $(1,2, \\\\ldots, n)$?')\n",
      "'No'\n",
      "('The problem asks whether a positive integer $n$ must exist such that the '\n",
      " 'given condition holds. This is a yes/no question, and the provided answer is '\n",
      " '\"no.\" This indicates the problem likely requires a proof to demonstrate the '\n",
      " 'existence or non-existence of such an $n$. Thus, the problem falls under '\n",
      " 'Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{2}$\\n')\n",
      "('Do there exist two bounded sequences $a_{1}, a_{2}, \\\\ldots$ and $b_{1}, '\n",
      " 'b_{2}, \\\\ldots$ such that for each positive integers $n$ and $m > n$ at '\n",
      " 'least one of the two inequalities $|a_{m} - a_{n}| > \\\\frac{1}{\\\\sqrt{n}}, '\n",
      " '|b_{m} - b_{n}| > \\\\frac{1}{\\\\sqrt{n}}$ holds?')\n",
      "'No, such sequences do not exist.'\n",
      "('The problem asks whether two sequences with a specific property exist.  '\n",
      " 'While the answer provided is \"no\", the core of the problem is a proof of '\n",
      " 'existence (or, in this case, non-existence). Thus, this falls under Case 2.\\n'\n",
      " '\\n'\n",
      " '[[2]]\\n')\n",
      "('If you must choose between selling Items 1 and 2 separately and selling them '\n",
      " 'in a bundle, which one do you choose? Is one strategy always better than the '\n",
      " 'other? Why?')\n",
      "'Neither strategy is always better'\n",
      "('The problem asks to compare two pricing strategies (selling items separately '\n",
      " 'versus bundling) and determine if one is always superior. The answer '\n",
      " 'provided states that \"neither strategy is always better\". This implies a '\n",
      " 'comparison and a conclusion, not a direct calculation or a proof of a '\n",
      " \"mathematical statement. This falls under Case 3 as it's closer to a business \"\n",
      " 'decision problem than a mathematical one.\\n'\n",
      " '\\n'\n",
      " '[[3]]\\n')\n",
      "('For a set $S$ of nonnegative integers, let $r_S(n)$ denote the number of '\n",
      " 'ordered pairs $(s_1, s_2)$ such that $s_1 \\\\in S$, $s_2 \\\\in S$, $s_1 \\\\ne '\n",
      " 's_2$, and $s_1 + s_2 = n$. Is it possible to partition the nonnegative '\n",
      " 'integers into two sets $A$ and $B$ in such a way that $r_A(n) = r_B(n)$ for '\n",
      " 'all $n$?')\n",
      "'Yes, such a partition is possible.'\n",
      "('The problem asks whether it is possible to partition the nonnegative '\n",
      " 'integers into two sets with a specific property. The answer provided is a '\n",
      " 'simple \"yes,\" indicating that such a partition exists. Although the problem '\n",
      " 'may involve a proof to show such a partition is possible, the question '\n",
      " 'itself is a yes/no question. This makes it fall under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{2}$\\n')\n",
      "('Let $S$ be a set of rational numbers such that\\n'\n",
      " '\\\\begin{enumerate}\\n'\n",
      " '\\\\item[(a)] $0 \\\\in S$;\\n'\n",
      " '\\\\item[(b)] If $x \\\\in S$ then $x+1\\\\in S$ and $x-1\\\\in S$; and\\n'\n",
      " '\\\\item[(c)] If $x\\\\in S$ and $x\\\\not\\\\in\\\\{0,1\\\\}$, then '\n",
      " '$\\\\frac{1}{x(x-1)}\\\\in S$.\\n'\n",
      " '\\\\end{enumerate}\\n'\n",
      " 'Must $S$ contain all rational numbers?')\n",
      "'No'\n",
      "('The problem asks \"Must $S$ contain all rational numbers?\". This is a yes/no '\n",
      " 'question, and the answer provided is \"No\". Since the problem presents a '\n",
      " 'conjecture and asks to prove or disprove it, this falls under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Assume that $(a_n)_{n\\\\geq 1}$ is an increasing sequence of positive real '\n",
      " 'numbers such that $\\\\lim a_n/n=0$.  Must there exist infinitely many '\n",
      " 'positive integers $n$ such that $a_{n-i}+a_{n+i}<2a_n$ for '\n",
      " '$i=1,2,\\\\ldots,n-1$?')\n",
      "'Yes, there must exist infinitely many such n.'\n",
      "('The problem asks \"Must there exist infinitely many positive integers $n$ '\n",
      " 'such that ...?\". This is a yes/no question and the answer provided also '\n",
      " 'starts with a \"Yes\". Since the problem requires a proof to answer this '\n",
      " 'question, this falls under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('A divisor $d$ of a positive integer $n$ is said to be a [i]close[/i] divisor '\n",
      " 'of $n$ if $\\\\sqrt{n}<d<2\\\\sqrt{n}$. Does there exist a positive integer with '\n",
      " 'exactly $2020$ close divisors?')\n",
      "'\\\\text{Yes, there exists a positive integer with exactly 2020 close divisors.}'\n",
      "('The problem asks whether there exists a positive integer with exactly 2020 '\n",
      " 'close divisors. The answer is a simple yes/no, which indicates that we are '\n",
      " 'trying to prove a statement. Therefore, this is Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Does there exist a prime number whose decimal representation is of the form '\n",
      " '$3811\\\\cdots11$ (that is, consisting of the digits $3$ and $8$ in that '\n",
      " 'order, followed by one or more digits $1$)?')\n",
      "'\\\\text{No}'\n",
      "('The problem asks whether a prime number exists in a specific format. While '\n",
      " 'the answer provided is simply \"No,\" the core of the problem is a question of '\n",
      " 'existence, which necessitates a proof to justify the answer. Thus, this '\n",
      " 'falls under Case 2.\\n'\n",
      " '\\n'\n",
      " '[[2]]\\n')\n",
      "('Determine if there exists a finite set $A$ of positive integers satisfying '\n",
      " 'the following condition: for each $a\\\\in{A}$ at least one of two numbers '\n",
      " '$2a$ and\\n'\n",
      " '$\\\\frac{a}{3}$ belongs to $A$.')\n",
      "'\\\\text{No}'\n",
      "('The problem asks to determine if a finite set exists satisfying a given '\n",
      " 'condition. While the answer provided is simply \"no\", the problem implicitly '\n",
      " 'asks for a proof of this statement. Thus, this falls under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: [[2]]\\n')\n",
      "('Does there exist an angle $ \\\\alpha\\\\in(0,\\\\pi/2)$ such that $ '\n",
      " '\\\\sin\\\\alpha$, $ \\\\cos\\\\alpha$, $ \\\\tan\\\\alpha$ and $ \\\\cot\\\\alpha$, taken '\n",
      " 'in some order, are consecutive terms of an arithmetic progression?')\n",
      "'\\\\text{No}'\n",
      "('The problem asks whether there exists an angle $\\\\alpha$ satisfying a '\n",
      " 'specific condition. While the answer is simply \"No,\" arriving at this answer '\n",
      " 'likely involves mathematical reasoning and potentially even a proof by '\n",
      " 'contradiction. Therefore, this falls under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is [[2]]\\n')\n",
      "('Is there an eight-digit number without zero digits, which when divided by '\n",
      " 'the first digit gives the remainder $1$, when divided by the second digit '\n",
      " 'will give the remainder $2$, ..., when divided by the eighth digit will give '\n",
      " 'the remainder $8$?')\n",
      "'\\\\text{No such number exists}'\n",
      "('The problem asks whether such a number exists. Although the answer states '\n",
      " '\"no\", the problem is asking to prove a conjecture. Thus, this is Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{2}$\\n')\n",
      "('Let $S = \\\\{2, 3, 4, \\\\ldots\\\\}$ denote the set of integers that are greater '\n",
      " 'than or equal to $2$. Does there exist a function $f : S \\\\to S$ such that '\n",
      " '\\\\[f (a)f (b) = f (a^2 b^2 )\\\\text{ for all }a, b \\\\in S\\\\text{ with }a \\\\ne '\n",
      " 'b?\\\\]')\n",
      "'\\\\text{No}'\n",
      "('The problem asks whether a function with a specific property exists. '\n",
      " 'Although the answer provided is \"No\", the core of the problem is about the '\n",
      " 'existence of such a function, which makes it a proof-based problem. Thus, '\n",
      " 'this falls under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: [[2]]\\n')\n",
      "('Can one find 4004 positive integers such that the sum of any 2003 of them is '\n",
      " 'not divisible by 2003?')\n",
      "'\\\\text{Yes}'\n",
      "('The problem asks a yes/no question, \"Can one find...\". The answer provided '\n",
      " 'is simply \"Yes.\" This suggests the problem likely requires a proof to '\n",
      " 'justify the \"Yes\" answer. Therefore, this falls under Case 2.\\n'\n",
      " '\\n'\n",
      " '[[2]]\\n')\n",
      "('Is there an integer $n$ such that $\\\\sqrt{n-1}+\\\\sqrt{n+1}$ is a rational '\n",
      " 'number?')\n",
      "'\\\\text{No}'\n",
      "('The problem asks whether there exists an integer  n such that the given '\n",
      " 'expression is rational.  This is a yes/no question, and the answer provided '\n",
      " 'requires a proof to show that such an n does not exist. Thus, this is Case '\n",
      " '2.\\n'\n",
      " '\\n'\n",
      " '[[2]]\\n')\n",
      "('Does there exist a function $f: \\\\mathbb R \\\\to \\\\mathbb R $  satisfying the '\n",
      " 'following conditions: \\n'\n",
      " '(i) for each real $y$  there is a real  $x$ such that $f(x)=y$ , and \\n'\n",
      " '(ii)  $f(f(x)) = (x - 1)f(x) + 2$ for all real $x$ ?')\n",
      "'\\\\text{No}'\n",
      "('The problem asks whether a function exists satisfying two given conditions. '\n",
      " 'Although the answer provided is \"No\", the core task is to determine the '\n",
      " 'existence of such a function. This makes it a proof-based problem focused on '\n",
      " 'establishing or refuting a claim, fitting the criteria for Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Let $a, b, c$ be non-negative numbers with $a+b+c = 3$. Prove the '\n",
      " 'inequality\\n'\n",
      " '\\\\[\\\\frac{a}{b^2+1}+\\\\frac{b}{c^2+1}+\\\\frac{c}{a^2+1} \\\\geq \\\\frac 32.\\\\]')\n",
      "'\\\\frac{3}{2}'\n",
      "('The problem asks to prove an inequality. This clearly falls under Case 2, '\n",
      " 'since we are asked to prove a statement. Although the answer is provided as '\n",
      " '3/2, the problem itself is asking for a proof, which makes it Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Two ants are moving along the edges of a convex polyhedron. The route of '\n",
      " 'every ant ends in its starting point, so that one ant does not pass through '\n",
      " 'the same point twice along its way. On every face $F$ of the polyhedron are '\n",
      " 'written the number of edges of $F$ belonging to the route of the first ant '\n",
      " 'and the number of edges of $F$ belonging to the route of the second ant. Is '\n",
      " 'there a polyhedron and a pair of routes described as above, such that only '\n",
      " 'one face contains a pair of distinct numbers?')\n",
      "'\\\\text{No}'\n",
      "('The problem asks whether a polyhedron and a pair of routes exist with the '\n",
      " 'given conditions. This is a yes/no question and the answer provided is '\n",
      " '\"No.\"  The problem requires a proof to show that such a polyhedron and '\n",
      " 'routes cannot exist. Thus, this falls under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Is there a quadratic trinomial with integer coefficients, such that all '\n",
      " 'values which are natural to be natural powers of two?')\n",
      "'\\\\text{No, there is no such quadratic trinomial.}'\n",
      "('The problem asks whether a specific type of quadratic trinomial exists. '\n",
      " 'Although the answer provided includes \"no,\" the core of the problem is about '\n",
      " 'the existence of such a trinomial, which implies a proof is required to '\n",
      " 'definitively answer the question. Thus, this falls under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: [[2]]\\n')\n",
      "('Arnim and Brentano have a little vase with $1500$ candies on the table and a '\n",
      " 'huge sack with spare candies under the table. They play a game taking turns, '\n",
      " 'Arnim begins . At each move a player can either eat $7$ candies or take $6$ '\n",
      " 'candies from under the table and add them to the vase. A player cannot go '\n",
      " 'under the table in two consecutive moves. A player is declared the winner if '\n",
      " 'he leaves the vase empty. In any other case, if a player cannot make a move '\n",
      " 'in his turn, the game is declared a tie. Is there a winning strategy for one '\n",
      " 'of the players?')\n",
      "'\\\\text{Brentano has a winning strategy.}'\n",
      "('The problem asks whether there exists a winning strategy for one of the '\n",
      " 'players. Although the answer states Brentano has a winning strategy, the '\n",
      " 'question itself is a yes/no question. Thus, this falls under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "'Is it possible to find $p,q,r\\\\in\\\\mathbb Q$ such that $p+q+r=0$ and $pqr=1$?'\n",
      "'\\\\text{No}'\n",
      "('The problem asks whether it is possible to find rational numbers $p, q, r$ '\n",
      " 'satisfying two equations. Although the answer provided is simply \"No\", the '\n",
      " 'problem itself is asking for specific values of $p, q, r$, or a proof if no '\n",
      " 'such values exist.  This makes it a problem with a yes/no answer that '\n",
      " 'requires a proof, which falls under Case 2.\\n'\n",
      " '\\n'\n",
      " '[[2]]\\n')\n",
      "('There are 20 buns with jam and 20 buns with treacle arranged in a row in '\n",
      " 'random order. Alice and Bob take in turn a bun from any end of the row. '\n",
      " 'Alice starts, and wants to finally obtain 10 buns of each type; Bob tries to '\n",
      " 'prevent this. Is it true for any order of the buns that Alice can win no '\n",
      " 'matter what are the actions of Bob?')\n",
      "'\\\\text{Yes}'\n",
      "('The problem asks \"Is it true for any order of the buns that Alice can win no '\n",
      " 'matter what are the actions of Bob?\". This is equivalent to asking to prove '\n",
      " 'that Alice can always win. Thus, this is a Case 2 problem.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('$40$ cells were marked on an infinite chessboard. Is it always possible to '\n",
      " 'find a rectangle that contains $20$ marked cells?\\n'\n",
      " '\\n'\n",
      " 'M. Evdokimov')\n",
      "'\\\\text{No}'\n",
      "('The problem asks a yes/no question, and the answer provided is simply \"no.\" '\n",
      " 'This strongly suggests the problem belongs to Case 2, requiring a proof to '\n",
      " 'justify the \"no\" answer.  The problem poses a question about the existence '\n",
      " 'of a rectangle satisfying a certain condition, which is a typical setup for '\n",
      " 'a proof-based problem.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Determine whether there exists an infinite sequence of nonzero digits $a_1 , '\n",
      " 'a_2 , a_3 , \\\\cdots $ and a positive integer $N$ such that for every integer '\n",
      " '$k > N$, the number $\\\\overline{a_k a_{k-1}\\\\cdots a_1 }$ is a perfect '\n",
      " 'square.')\n",
      "'\\\\text{No}'\n",
      "('The problem asks to determine whether such a sequence and integer exist. '\n",
      " 'Although the answer is a simple \"no,\" the problem itself requires a proof to '\n",
      " 'show that no such sequence and integer exist. Therefore, this is a Case 2 '\n",
      " 'problem.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{2}$\\n')\n",
      "('Does there exist a positive integer $ n$ such that $ n$ has exactly 2000 '\n",
      " 'prime divisors and $ n$ divides $ 2^n \\\\plus{} 1$?')\n",
      "'\\\\text{Yes}'\n",
      "('The problem asks whether a positive integer $n$ exists with certain '\n",
      " 'properties. Although the answer provided is simply \"Yes\", the problem itself '\n",
      " 'is asking for the existence of such an integer, which implies a need for '\n",
      " 'proof to justify the \"Yes\" answer. Therefore, this falls under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('We call a two-variable polynomial $P(x, y)$ [i]secretly one-variable,[/i] if '\n",
      " 'there exist polynomials $Q(x)$ and $R(x, y)$ such that $\\\\deg(Q) \\\\ge 2$ and '\n",
      " '$P(x, y) = Q(R(x, y))$ (e.g. $x^2 + 1$ and $x^2y^2 +1$ are [i]secretly '\n",
      " 'one-variable[/i], but $xy + 1$ is not).\\n'\n",
      " '\\n'\n",
      " 'Prove or disprove the following statement: If $P(x, y)$ is a polynomial such '\n",
      " 'that both $P(x, y)$ and $P(x, y) + 1$ can be written as the product of two '\n",
      " 'non-constant polynomials, then $P$ is [i]secretly one-variable[/i].')\n",
      "'\\\\text{True}'\n",
      "('The problem asks to prove or disprove a statement. This clearly falls under '\n",
      " 'Case 2. Although the answer is given as True, the core task is to construct '\n",
      " 'a proof, which aligns with the characteristics of Case 2 problems.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Determine whether or not there exist 15 integers $m_1,\\\\ldots,m_{15}$\\n'\n",
      " '  such that~\\n'\n",
      " '  $$\\\\displaystyle \\\\sum_{k=1}^{15}\\\\,m_k\\\\cdot\\\\arctan(k) = \\\\arctan(16). '\n",
      " '\\\\eqno(1)$$\\n'\n",
      " '\\n'\n",
      " '(')\n",
      "'\\\\text{No}'\n",
      "('The problem asks to determine whether or not there exist 15 integers such '\n",
      " 'that a given equation holds. This is equivalent to asking to prove or '\n",
      " 'disprove the existence of these integers. Thus, this problem falls under '\n",
      " 'Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: [[2]]\\n')\n",
      "('Let $ABC$ be a triangle with $\\\\angle A = 90^{\\\\circ}$. Points $D$ and $E$ '\n",
      " 'lie on sides $AC$ and $AB$, respectively, such that $\\\\angle ABD = \\\\angle '\n",
      " 'DBC$ and $\\\\angle ACE = \\\\angle ECB$. Segments $BD$ and $CE$ meet at $I$. '\n",
      " 'Determine whether or not it is possible for segments $AB$, $AC$, $BI$, $ID$, '\n",
      " '$CI$, $IE$ to all have integer lengths.')\n",
      "'\\\\text{No}'\n",
      "(\"The problem asks to determine whether or not it's possible for certain \"\n",
      " 'segments to have integer lengths. This is equivalent to asking \"does there '\n",
      " 'exist a configuration such that $AB, AC, BI, ID, CI, IE$ are all integers?\". '\n",
      " 'Thus, the problem seeks a yes/no answer, which makes it a proof-based '\n",
      " 'problem. Although the answer is provided as \"No\", the core of the question '\n",
      " 'lies in proving this statement. Therefore, this is a Case 2 problem.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Let $a,b,c,d$ be non-negative reals such that $a+b+c+d=4$. Prove the '\n",
      " 'inequality\\n'\n",
      " '\\\\[\\\\frac{a}{a^3+8}+\\\\frac{b}{b^3+8}+\\\\frac{c}{c^3+8}+\\\\frac{d}{d^3+8}\\\\le\\\\frac{4}{9}\\\\]')\n",
      "('\\\\frac{a}{a^3 + 8} + \\\\frac{b}{b^3 + 8} + \\\\frac{c}{c^3 + 8} + \\\\frac{d}{d^3 '\n",
      " '+ 8} \\\\leq \\\\frac{4}{9}')\n",
      "('The problem asks to prove an inequality. This clearly falls under Case 2, '\n",
      " 'which involves proving or disproving a statement.  Even if a solution '\n",
      " 'exists, the core of the problem lies in constructing a proof to validate the '\n",
      " 'given inequality.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: [[2]]\\n')\n",
      "('There are $n > 2022$ cities in the country. Some pairs of cities are '\n",
      " 'connected with straight two-ways airlines. Call the set of the cities {\\\\it '\n",
      " 'unlucky}, if it is impossible to color the airlines between them in two '\n",
      " 'colors without monochromatic triangle (i.e. three cities $A$, $B$, $C$ with '\n",
      " 'the airlines $AB$, $AC$ and $BC$ of the same color). \\n'\n",
      " '\\n'\n",
      " 'The set containing all the cities is unlucky. Is there always an unlucky set '\n",
      " 'containing exactly 2022 cities?')\n",
      "'\\\\text{No}'\n",
      "('The problem asks if there always exists an unlucky set containing exactly '\n",
      " '2022 cities, given that the set of all cities is unlucky. The answer is no.  '\n",
      " 'This problem is asking to prove or disprove the existence of a certain '\n",
      " 'configuration. This clearly falls under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is [[2]]\\n')\n",
      "('A sequence $(a_n)_{n=1}^{\\\\infty}$ of positive integers satisfies the '\n",
      " 'condition $a_{n+1} = a_n +\\\\tau (n)$ for all positive integers $n$ where '\n",
      " '$\\\\tau (n)$ is the number of positive integer divisors of $n$. Determine '\n",
      " 'whether two consecutive terms of this sequence can be perfect squares.')\n",
      "'\\\\text{No}'\n",
      "('The problem asks to determine whether two consecutive terms can be perfect '\n",
      " 'squares.  The answer provided is a simple \"No\". This indicates that the '\n",
      " 'problem is asking for a proof to a yes/no question rather than finding a '\n",
      " 'specific value or set of values. This falls under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Seyed has 998 white coins, a red coin, and an unusual coin with one red side '\n",
      " 'and one white side. He can not see the color of the coins instead he has a '\n",
      " 'scanner which checks if all of the coin sides touching the scanner glass are '\n",
      " 'white. Is there any algorithm to find the red coin by using the scanner at '\n",
      " 'most 17 times?')\n",
      "'\\\\text{YES}'\n",
      "('The problem asks for an algorithm (\"Is there any algorithm...\") and the '\n",
      " 'answer is YES. This implies the problem is looking for a constructive proof '\n",
      " 'of existence, which falls under Case 2.\\n'\n",
      " '\\n'\n",
      " '[[2]]\\n')\n",
      "('An [i]anti-Pascal[/i] triangle is an equilateral triangular array of numbers '\n",
      " 'such that, except for the numbers in the bottom row, each number is the '\n",
      " 'absolute value of the difference of the two numbers immediately below it. '\n",
      " 'For example, the following is an anti-Pascal triangle with four rows which '\n",
      " 'contains every integer from $1$ to $10$.\\n'\n",
      " '\\\\[\\\\begin{array}{\\n'\n",
      " 'c@{\\\\hspace{4pt}}c@{\\\\hspace{4pt}}\\n'\n",
      " 'c@{\\\\hspace{4pt}}c@{\\\\hspace{2pt}}c@{\\\\hspace{2pt}}c@{\\\\hspace{4pt}}c\\n'\n",
      " '} \\\\vspace{4pt}\\n'\n",
      " ' & & & 4 & & &  \\\\\\\\\\\\vspace{4pt}\\n'\n",
      " ' & & 2 & & 6 & &  \\\\\\\\\\\\vspace{4pt}\\n'\n",
      " ' & 5 & & 7 & & 1 & \\\\\\\\\\\\vspace{4pt}\\n'\n",
      " ' 8 & & 3 & & 10 & & 9 \\\\\\\\\\\\vspace{4pt}\\n'\n",
      " '\\\\end{array}\\\\]\\n'\n",
      " 'Does there exist an anti-Pascal triangle with $2018$ rows which contains '\n",
      " 'every integer from $1$ to $1 + 2 + 3 + \\\\dots + 2018$?\\n'\n",
      " '\\n'\n",
      " '[i]')\n",
      "'\\\\text{No}'\n",
      "('The problem asks \"Does there exist an anti-Pascal triangle...\" and the '\n",
      " 'answer provided is \"No\". This indicates that the problem requires proving or '\n",
      " 'disproving the existence of such a triangle. Therefore, it falls under Case '\n",
      " '2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: [[2]]\\n')\n",
      "('Does there exist a sequence $ F(1), F(2), F(3), \\\\ldots$ of non-negative '\n",
      " 'integers that simultaneously satisfies the following three conditions?\\n'\n",
      " '\\n'\n",
      " '[b](a)[/b] Each of the integers $ 0, 1, 2, \\\\ldots$ occurs in the sequence.\\n'\n",
      " '[b](b)[/b] Each positive integer occurs in the sequence infinitely often.\\n'\n",
      " '[b](c)[/b] For any $ n \\\\geq 2,$\\n'\n",
      " '\\\\[ F(F(n^{163})) \\\\equal{} F(F(n)) \\\\plus{} F(F(361)).\\n'\n",
      " '\\\\]')\n",
      "'\\\\text{Yes}'\n",
      "('The problem asks whether a sequence with certain properties exists. While '\n",
      " 'the answer provided is simply \"yes,\" the problem itself is asking to prove '\n",
      " 'the existence of such a sequence. Thus, it falls under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{2}$\\n')\n",
      "('Does there exist positive integers $n_1, n_2, \\\\dots, n_{2022}$ such that '\n",
      " 'the number\\n'\n",
      " '$$\\n'\n",
      " '  \\\\left( n_1^{2020} + n_2^{2019} \\\\right)\\\\left( n_2^{2020} + n_3^{2019} '\n",
      " '\\\\right) \\\\cdots \\\\left( n_{2021}^{2020} + n_{2022}^{2019} \\\\right)\\\\left( '\n",
      " 'n_{2022}^{2020} + n_1^{2019} \\\\right)\\n'\n",
      " '$$\\n'\n",
      " 'is a power of $11$?')\n",
      "'\\\\text{No}'\n",
      "('The problem asks whether there exist positive integers satisfying a given '\n",
      " 'condition.  The answer is a simple \"No\".  Thus, this is a Case 2 problem.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Does there exist a sequence of positive integers $a_1,a_2,...$ such that '\n",
      " 'every positive integer occurs exactly once and that the number $\\\\tau '\n",
      " '(na_{n+1}^n+(n+1)a_n^{n+1})$ is divisible by $n$ for all positive integer.\\n'\n",
      " '\\n'\n",
      " 'Here $\\\\tau (n)$ denotes the number of positive divisor of $n$.')\n",
      "'$  \\\\text{ Yes }$'\n",
      "('The problem asks whether a sequence with a specific property exists.  While '\n",
      " 'the answer provided is \"yes,\" the problem itself is about the existence of '\n",
      " 'such a sequence, which implies a need for proof. This falls under Case 2.\\n'\n",
      " '\\n'\n",
      " '[[2]]\\n')\n",
      "('Does there exist a finite set of real numbers such that their sum equals '\n",
      " '$2$, the sum of their squares equals $3$, the sum of their cubes equals $4$, '\n",
      " '..., and the sum of their ninth powers equals $10$?')\n",
      "'\\\\text{no}'\n",
      "('The problem asks whether a finite set of real numbers exists satisfying a '\n",
      " 'given set of conditions. The answer is a direct \"no.\" Thus, this falls under '\n",
      " 'Case 2, where a proof is required to show that such a set does not exist.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{2}$\\n')\n",
      "('For every positive integer $n$, let $f(n)$, $g(n)$ be the minimal positive '\n",
      " 'integers such that\\n'\n",
      " '\\\\[1+\\\\frac{1}{1!}+\\\\frac{1}{2!}+\\\\dots '\n",
      " '+\\\\frac{1}{n!}=\\\\frac{f(n)}{g(n)}.\\\\]\\n'\n",
      " 'Determine whether there exists a positive integer $n$ for which '\n",
      " '$g(n)>n^{0.999n}$.')\n",
      "'\\\\text{yes}'\n",
      "('The problem asks to determine whether there exists a positive integer  n for '\n",
      " 'which \\\\(g(n) > n^{0.999n}\\\\).  Since the problem is a yes/no question, it '\n",
      " 'falls under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: [[2]]\\n')\n",
      "('Each of the six boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$, $B_6$ initially '\n",
      " 'contains one coin. The following operations are allowed\\n'\n",
      " '\\n'\n",
      " 'Type 1) Choose a non-empty box $B_j$, $1\\\\leq j \\\\leq 5$, remove one coin '\n",
      " 'from $B_j$ and add two coins to $B_{j+1}$; \\n'\n",
      " '\\n'\n",
      " 'Type 2) Choose a non-empty box $B_k$, $1\\\\leq k \\\\leq 4$, remove one coin '\n",
      " 'from $B_k$ and swap the contents (maybe empty) of the boxes $B_{k+1}$ and '\n",
      " '$B_{k+2}$.\\n'\n",
      " '\\n'\n",
      " 'Determine if there exists a finite sequence of operations of the allowed '\n",
      " 'types, such that the five boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$ become '\n",
      " 'empty, while box $B_6$ contains exactly $2010^{2010^{2010}}$ coins.\\n'\n",
      " '\\n'\n",
      " '[i]')\n",
      "'$\\\\text{No}$'\n",
      "('The problem asks to determine whether a certain configuration can be reached '\n",
      " 'using given operations. The answer provided is \"No.\"  This means the problem '\n",
      " 'seeks a definitive yes/no answer. The problem is therefore a proof-based '\n",
      " 'problem.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Let $a, b, c$ be  positive real numbers such that $abc = \\\\dfrac {1} {8}$. '\n",
      " 'Prove the inequality:$$a ^ 2 + b ^ 2 + c ^ 2 + a ^ 2b ^ 2 + b ^ 2c ^ 2 + c ^ '\n",
      " '2a ^ 2 \\\\geq \\\\dfrac {15} {16}$$\\n'\n",
      " 'When the equality holds?')\n",
      "'a=b=c=\\\\dfrac{1}{2}'\n",
      "('The problem asks to prove an inequality. This clearly falls under Case 2. '\n",
      " 'Although there is an answer provided for when equality holds, the core of '\n",
      " 'the problem is proving the inequality.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('For a positive integer $n$, let $d(n)$ be the number of positive divisors of '\n",
      " '$n$, and let $\\\\varphi(n)$ be the number of positive integers not exceeding '\n",
      " '$n$ which are coprime to $n$. Does there exist a constant $C$ such that\\n'\n",
      " '\\n'\n",
      " '$$ \\\\frac {\\\\varphi ( d(n))}{d(\\\\varphi(n))}\\\\le C$$\\n'\n",
      " 'for all $n\\\\ge 1$\\n'\n",
      " '\\n'\n",
      " '[i]Cyprus[/i]')\n",
      "'$\\\\text{ No }$'\n",
      "('The problem asks whether there exists a constant $C$ such that an inequality '\n",
      " 'holds for all $n \\\\ge 1$.  This is a yes/no question, and the answer '\n",
      " 'provided confirms this by stating \"No\". The problem requires a proof to show '\n",
      " 'why no such constant exists. Thus, this falls under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Does there exist a set $M$ in usual Euclidean space such that for every '\n",
      " 'plane $\\\\lambda$ the intersection $M \\\\cap \\\\lambda$ is finite and nonempty '\n",
      " '?\\n'\n",
      " '\\n'\n",
      " '[i]\\n'\n",
      " '[hide=\"Remark\"]I\\'m not sure I\\'m posting this in a right Forum.[/hide]')\n",
      "'\\\\text{yes}'\n",
      "('The problem asks \"Does there exist...\" which implies a yes/no answer.  The '\n",
      " 'problem aims to prove the existence of a set with a specific property for '\n",
      " 'its intersection with any plane. The answer provided is simply \"yes\". This '\n",
      " 'indicates that the problem seeks a proof of existence, which falls under '\n",
      " 'Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Is there a number $n$ such that one can write $n$ as the sum of $2017$ '\n",
      " 'perfect squares and (with at least) $2017$ distinct ways?')\n",
      "'\\\\text{Yes}'\n",
      "('The problem asks whether there exists a number $n$ with a specific property. '\n",
      " 'Although the answer provided is simply \"Yes\", the problem itself is a '\n",
      " 'question of existence.  This falls under Case 2, problems that require a '\n",
      " 'proof to answer Yes/No.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('A palindrome is a positive integer which reads in the same way in both '\n",
      " 'directions (for example, $1$, $343$ and $2002$ are palindromes, while $2005$ '\n",
      " 'is not). Is it possible to find $2005$ pairs in the form of $(n, n + 110)$ '\n",
      " 'where both numbers are palindromes?')\n",
      "'\\\\text{Yes}'\n",
      "('The problem asks if it is possible to find 2005 pairs of a specific form '\n",
      " 'where both numbers are palindromes.  The answer is a simple \"yes\". This '\n",
      " 'problem is asking for a proof of existence, and thus falls under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is [[2]]\\n')\n",
      "('Does there exist a function $s\\\\colon \\\\mathbb{Q} \\\\rightarrow \\\\{-1,1\\\\}$ '\n",
      " 'such that if $x$ and $y$ are distinct rational numbers satisfying ${xy=1}$ '\n",
      " 'or ${x+y\\\\in \\\\{0,1\\\\}}$, then ${s(x)s(y)=-1}$? Justify your answer.\\n'\n",
      " '\\n'\n",
      " '[i]')\n",
      "'\\\\text{Yes}'\n",
      "('The problem asks whether a function with a specific property exists. While '\n",
      " 'the problem mentions \"Justify your answer\", the core of the problem is to '\n",
      " 'determine the existence of such a function, which is a yes/no question. '\n",
      " 'Since it asks to justify the answer, it falls under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('We know that the orthocenter reflects over the sides of the triangle on the '\n",
      " 'circumcircle. Therefore the minimal distance $ OD\\\\plus{}HD$ equals $ R$. '\n",
      " 'Obviously we can achieve this on all sides, so we assume that $ D,E,F$ are '\n",
      " \"the intersection points between $ A',B',C'$ the reflections of $ H$ across $ \"\n",
      " 'BC,CA,AB$ respectively. All we have to prove is that $ AD$, $ BE$ and $ CF$ '\n",
      " 'are concurrent. \\n'\n",
      " '\\n'\n",
      " 'In order to do that we need the ratios $ \\\\dfrac {BD}{DC}$, $ \\\\dfrac '\n",
      " \"{CE}{EA}$ and $ \\\\dfrac {AF}{FB}$, and then we can apply Ceva's theorem. \\n\"\n",
      " '\\n'\n",
      " 'We know that the triangle $ ABC$ is acute, so $ \\\\angle BAH \\\\equal{} '\n",
      " '90^\\\\circ\\\\minus{} \\\\angle B \\\\equal{} \\\\angle OAC$, therefore $ \\\\angle HAO '\n",
      " '\\\\equal{} |\\\\angle A \\\\minus{} 2(90^\\\\circ \\\\minus{}\\\\angle B)| \\\\equal{} '\n",
      " '|\\\\angle B\\\\minus{} \\\\angle C|$. In particular this means that $ \\\\angle '\n",
      " \"OA'H \\\\equal{} |\\\\angle B\\\\minus{}\\\\angle C|$. Since $ \\\\angle BA'A \"\n",
      " \"\\\\equal{} \\\\angle C$ and $ \\\\angle AA'C \\\\equal{} \\\\angle B$, we have that $ \"\n",
      " \"\\\\angle BA'D \\\\equal{} \\\\angle B$ and $ \\\\angle DA'C \\\\equal{} \\\\angle C$.\\n\"\n",
      " '\\n'\n",
      " \"By the Sine theorem in the triangles $ BA'D$ and $ DA'C$, we get \\n\"\n",
      " '\\\\[ \\\\dfrac {BD}{DC} \\\\equal{} \\\\dfrac { \\\\sin B }{\\\\sin C }.\\\\] \\n'\n",
      " '\\n'\n",
      " 'Using the similar relationships for  $ \\\\dfrac {CE}{EA}$ and $ \\\\dfrac '\n",
      " '{AF}{FB}$ we get that those three fractions multiply up to 1, and thus by '\n",
      " \"Ceva's, the lines $ AD, BE$ and $ CF$ are concurrent.\")\n",
      "''\n",
      "('The problem asks to prove that $AD$, $BE$, and $CF$ are concurrent. This is '\n",
      " 'a proof-based problem, fitting into Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{2}$\\n')\n",
      "(\"Let's assume $x,y>0$ (clearly, we can do this, since if what we want to \"\n",
      " \"prove doesn't hold, then it doesn't hold if we replace $x$ with $-x$ and/or \"\n",
      " \"$y$ with $-y$). Let's work with non-negative integers only.\\n\"\n",
      " '\\n'\n",
      " 'The negation of what we want to prove states that there is a set $S\\\\subset '\n",
      " '\\\\mathbb N$ s.t. $S,S+x,S+y,S+x+y$ are mutually disjoint, and their union is '\n",
      " '$\\\\mathbb N$. This means that, working with formal power series, '\n",
      " '$1+t+t^2+\\\\ldots=\\\\left(\\\\sum_{s\\\\in S}t^s\\\\right)(1+t^x)(1+t^y)$. Assume '\n",
      " 'now that $y<x$. We have '\n",
      " '$\\\\frac{1+t+t^2+\\\\ldots}{1+t^y}=(1+t+\\\\ldots+t^{y-1})+(t^{2y}+t^{2y+1}+\\\\ldots+t^{3y-1})+\\\\ldots=\\\\mathcal '\n",
      " 'E$. \\n'\n",
      " '\\n'\n",
      " 'When we divide $\\\\mathcal E$ by $1+t^x$ we have to get a series whose only '\n",
      " 'coefficients are $0$ and $1$, and this will yield the contradiction: our '\n",
      " 'series contains $1+t+\\\\ldots+t^{y-1}$, because $y<x$. There must be a $k$ '\n",
      " 's.t. $x\\\\in(2ky,(2k+1)y-1)$ (the interval is open because the endpoints are '\n",
      " 'even, but $x$ is odd). However, there is an $\\\\alpha\\\\in\\\\overline{0,y-1}$ '\n",
      " 's.t. $x+\\\\alpha=(2k+1)y$, and this means that if our power series has no '\n",
      " 'negative terms (to get rid of $t^{(2k+1)y}$, which does not appear in '\n",
      " '$\\\\mathcal E$), when multiplied by $1+t^x$ contains $t^{(2k+1)y}$, but '\n",
      " \"$\\\\mathcal E$ doesn't have this term, so we have a contradiction.\")\n",
      "''\n",
      "('The problem states \"Let\\'s assume $x,y>0$ ... The negation of what we want '\n",
      " 'to prove states that there is a set...\". This indicates the problem aims to '\n",
      " 'prove a statement by contradiction.  The answer provides a detailed argument '\n",
      " 'using formal power series and intervals to arrive at a contradiction. Thus, '\n",
      " 'the problem seeks a proof.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: [[2]]\\n')\n",
      "('For a triangle $ ABC,$ let $ k$ be its circumcircle with radius $ r.$ The '\n",
      " 'bisectors of the inner angles $ A, B,$ and $ C$ of the triangle intersect '\n",
      " \"respectively the circle $ k$ again at points $ A', B',$ and $ C'.$ Prove the \"\n",
      " 'inequality\\n'\n",
      " '\\n'\n",
      " '\\\\[ 16Q^3 \\\\geq 27 r^4 P,\\\\]\\n'\n",
      " '\\n'\n",
      " \"where $ Q$ and $ P$ are the areas of the triangles $ A'B'C'$ and $ABC$ \"\n",
      " 'respectively.')\n",
      "'Q^3\\\\geq\\\\frac{27}{16}r^4P\\\\Leftrightarrow16Q^3\\\\geq27r^4P'\n",
      "('The problem asks to prove an inequality relating the areas of triangle '\n",
      " \"A'B'C' and triangle ABC, where A', B', and C' are points on the circumcircle \"\n",
      " 'of triangle ABC. Although the problem statement says \"prove\", its nature is '\n",
      " 'to establish the given inequality. This is different from problems that ask '\n",
      " 'for a proof to answer Yes/No or prove/disprove a statement (Case 2). '\n",
      " 'Therefore, this problem falls under Case 1.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: [[2]]\\n')\n",
      "('Does there exist a set $S$ of $100$ points in a plane such that the center '\n",
      " 'of mass of any $10$ points in $S$ is also a point in $S$?')\n",
      "'\\\\text{No}'\n",
      "('The problem asks whether a certain type of set exists. Although the answer '\n",
      " 'provided is simply \"No\", the problem requires a proof to justify this '\n",
      " 'answer.  This makes it fall under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Do there exist two bounded sequences $a_1, a_2,\\\\ldots$ and $b_1, '\n",
      " 'b_2,\\\\ldots$ such that for each positive integers $n$  and $m>n$ at least '\n",
      " 'one of the two inequalities $|a_m-a_n|>1/\\\\sqrt{n},$ and '\n",
      " '$|b_m-b_n|>1/\\\\sqrt{n}$ holds?')\n",
      "'\\\\text{No}'\n",
      "('The problem asks whether two bounded sequences with a specific property '\n",
      " 'exist.  While the answer provided is simply \"No\", this indicates that the '\n",
      " 'problem is asking for a proof demonstrating the non-existence of such '\n",
      " 'sequences. Therefore, this falls under Case 2.\\n'\n",
      " '\\n'\n",
      " '[[2]]\\n')\n",
      "('There are $n>2$ lines on the plane in general position; Meaning any two of '\n",
      " 'them meet, but no three are concurrent. All their intersection points are '\n",
      " 'marked, and then all the lines are removed, but the marked points are '\n",
      " 'remained. It is not known which marked point belongs to which two lines. Is '\n",
      " 'it possible to know which line belongs where, and restore them all?')\n",
      "'\\\\text{Yes}'\n",
      "('The problem asks if it is possible to determine the original lines given the '\n",
      " 'intersection points. The answer provided is a simple \"Yes\".  This suggests '\n",
      " 'the problem is asking for a proof of the existence of a method to '\n",
      " 'reconstruct the lines.  This falls under Case 2 since it requires a proof.\\n'\n",
      " '\\n'\n",
      " '[[2]]\\n')\n",
      "('Prove: If the sum of all positive divisors of $ n \\\\in '\n",
      " '\\\\mathbb{Z}^{\\\\plus{}}$ is a power of two, then the number/amount of the '\n",
      " 'divisors is a power of two.')\n",
      "'\\\\text{The number of divisors is a power of two.}'\n",
      "('The problem asks to prove a statement.  The problem is a Case 2 since it '\n",
      " 'requires a proof to show whether the statement is true or false.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('A square grid on the Euclidean plane consists of all points $(m,n)$, where '\n",
      " '$m$ and $n$ are integers. Is it possible to cover all grid points by an '\n",
      " 'infinite family of discs with non-overlapping interiors if each disc in the '\n",
      " 'family has radius at least $5$?')\n",
      "'\\\\text{No, it is not possible to cover all grid points with such discs.}'\n",
      "('The problem asks whether it is possible to cover all grid points with discs '\n",
      " 'of radius at least 5. The answer is a direct \"No\". This falls under Case 2, '\n",
      " 'as the problem is asking to prove/disprove a statement.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{2}$\\n')\n",
      "('Let $S$ be the set of positive integers. For any $a$ and $b$ in the set we '\n",
      " 'have $GCD(a, b)>1$. For any $a$, $b$ and $c$ in the set we have $GCD(a, b, '\n",
      " 'c)=1$. Is it possible that $S$ has $2012$ elements?  \\n'\n",
      " '\\t\\t\\n'\n",
      " '[i]')\n",
      "'\\\\text{Yes}'\n",
      "('The problem asks whether it is possible for a set S to have 2012 elements '\n",
      " 'with the given GCD properties. This is a yes/no question that requires a '\n",
      " 'proof. Therefore, it falls under Case 2.\\n'\n",
      " '\\n'\n",
      " '[[2]]\\n')\n",
      "('Can one place positive integers at all vertices of a cube in such a way that '\n",
      " 'for every pair of numbers connected by an edge, one will be divisible by the '\n",
      " 'other , and there are no other pairs of numbers with this property? \\n'\n",
      " '\\n'\n",
      " '(A Shapovalov)')\n",
      "'\\\\text{Yes}'\n",
      "(\"The problem asks whether it's possible to arrange numbers on the vertices of \"\n",
      " 'a cube such that divisibility occurs only along edges.  The answer is a '\n",
      " 'simple \"yes,\" indicating a construction is possible. This is a classic '\n",
      " 'existence problem, and since the answer is \"Yes\", it implies there exists a '\n",
      " 'direct construction to demonstrate such an arrangement. This makes it a Case '\n",
      " '2 problem disguised as a Case 1.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: [[2]]\\n')\n",
      "('4. There are several (at least two) positive integers written along the '\n",
      " 'circle. For any two neighboring integers one is either twice as big as the '\n",
      " 'other or five times as big as the other. Can the sum of all these integers '\n",
      " 'equal 2023 ?\\n'\n",
      " '\\t\\n'\n",
      " '\\tSergey Dvoryaninov')\n",
      "'\\\\text{No}'\n",
      "('The problem asks whether a configuration of numbers exists under given '\n",
      " 'conditions.  This is a yes/no question and therefore falls under Case 2.\\n'\n",
      " '\\n'\n",
      " '[[2]]\\n')\n",
      "('A convex figure $F$ is such that any equilateral triangle with side $1$ has '\n",
      " 'a parallel translation that takes all its vertices to the boundary of $F$. '\n",
      " 'Is $F$ necessarily a circle?')\n",
      "'\\\\text{No}'\n",
      "('The problem asks whether a convex figure $F$ with a specific property is '\n",
      " 'necessarily a circle. The answer provided is \"No\". This problem presents a '\n",
      " 'statement and asks for a proof/disproof (whether the statement is '\n",
      " 'true/false). Therefore, this falls under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "(\"Can the 'brick wall' (infinite in all directions) drawn at the picture be \"\n",
      " 'made of wires of length $1, 2, 3, \\\\dots$ (each positive integral length  '\n",
      " 'occurs exactly once)? (Wires can be bent but should not overlap; size of a '\n",
      " \"'brick' is $1\\\\times 2$).\\n\"\n",
      " '\\n'\n",
      " '[asy]\\n'\n",
      " 'unitsize(0.5 cm);\\n'\n",
      " '\\n'\n",
      " 'for(int i = 1; i <= 9; ++i) {\\n'\n",
      " '  draw((0,i)--(10,i));\\n'\n",
      " '}\\n'\n",
      " '\\n'\n",
      " 'for(int i = 0; i <= 4; ++i) {\\n'\n",
      " '  for(int j = 0; j <= 4; ++j) {\\n'\n",
      " '    draw((2*i + 1,2*j)--(2*i + 1,2*j + 1));\\n'\n",
      " '  }\\n'\n",
      " '}\\n'\n",
      " '\\n'\n",
      " 'for(int i = 0; i <= 3; ++i) {\\n'\n",
      " '  for(int j = 0; j <= 4; ++j) {\\n'\n",
      " '    draw((2*i + 2,2*j + 1)--(2*i + 2,2*j + 2));\\n'\n",
      " '  }\\n'\n",
      " '}\\n'\n",
      " '[/asy]')\n",
      "'\\\\text{No, it is not possible.}'\n",
      "('The problem asks whether the brick wall can be made of wires of length $1, '\n",
      " '2, 3, \\\\dots$. The answer provided is \"No\". This indicates that the problem '\n",
      " 'is asking for a proof to a yes/no question. Thus, this problem falls under '\n",
      " 'Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('A number is written in each corner of the cube. On each step, each number is '\n",
      " 'replaced with the average of three numbers in the three adjacent corners '\n",
      " '(all the numbers are replaced simultaneously). After ten such steps, every '\n",
      " 'number returns to its initial value. Must all numbers have been originally '\n",
      " 'equal?')\n",
      "'\\\\text{No}'\n",
      "('The problem asks whether all the numbers must have been originally equal '\n",
      " 'given a certain condition. This is a yes/no question, and the answer '\n",
      " 'provided is \"no\". Since the problem aims to determine whether a statement is '\n",
      " 'necessarily true, and not to find a specific value or set of values, this is '\n",
      " 'a Case 2 problem.\\n'\n",
      " '\\n'\n",
      " '[[2]]\\n')\n",
      "('Amy and Bob play the game. At the beginning, Amy writes down a positive '\n",
      " 'integer on the board. Then the players take moves in turn, Bob moves first. '\n",
      " 'On any move of his, Bob replaces the number $n$ on the blackboard with a '\n",
      " 'number of the form $n-a^2$, where $a$ is a positive integer. On any move of '\n",
      " 'hers, Amy replaces the number $n$ on the blackboard with a number of the '\n",
      " 'form $n^k$, where $k$ is a positive integer. Bob wins if the number on the '\n",
      " 'board becomes zero.\\n'\n",
      " 'Can Amy prevent Bob’s win?')\n",
      "'\\\\text{No, Amy cannot prevent Bob from winning.}'\n",
      "(\"The problem asks whether Amy can prevent Bob's win. This is equivalent to \"\n",
      " 'asking if Bob has a winning strategy, which is a yes/no question, and the '\n",
      " \"answer provided confirms this by stating that Amy cannot prevent Bob's win, \"\n",
      " 'implying Bob has a winning strategy. This is a proof-based problem.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Does there exist an integer such that its cube is equal to $3n^2 + 3n + 7,$ '\n",
      " 'where $n$ is an integer.')\n",
      "'\\\\text{No}'\n",
      "('The problem asks whether there exists an integer whose cube can be expressed '\n",
      " 'in the form $3n^2 + 3n + 7$.  The answer is a simple \"No.\" This problem aims '\n",
      " 'to prove or disprove the existence of such an integer, which aligns with the '\n",
      " 'characteristics of a Case 2 problem. Although the answer is a direct \"No,\" '\n",
      " 'the core of the problem lies in proving this statement.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('We define two types of operation on polynomial of third degree:\\n'\n",
      " 'a) switch places of the coefficients of polynomial(including zero '\n",
      " 'coefficients), ex:\\n'\n",
      " '$ x^3+x^2+3x-2 $ => $ -2x^3+3x^2+x+1$\\n'\n",
      " 'b) replace the polynomial $P(x)$ with $P(x+1)$\\n'\n",
      " 'If limitless amount of operations is allowed,\\n'\n",
      " 'is it possible from $x^3-2$ to get $x^3-3x^2+3x-3$ ?')\n",
      "'\\\\text{No}'\n",
      "('The problem asks whether it is possible to transform a given polynomial into '\n",
      " 'another polynomial using specific operations.  While the answer provided is '\n",
      " 'simply \"No\", the question is fundamentally asking for the existence of a '\n",
      " 'sequence of transformations. Thus, it is a proof-based problem.\\n'\n",
      " '\\n'\n",
      " '[[2]]\\n')\n",
      "('A natural number is written in each cell of an $8 \\\\times 8$ board. It '\n",
      " 'turned out that for any tiling of the board with dominoes, the sum of '\n",
      " 'numbers in the cells of each domino is different. Can it happen that the '\n",
      " 'largest number on the board is no greater than $32$?')\n",
      "'\\\\text{Yes}'\n",
      "(\"The problem asks if it's possible for the largest number on the board to be \"\n",
      " 'no greater than 32 given a specific condition. The answer provided is '\n",
      " '\"Yes\".  This indicates a problem where a construction or counterexample '\n",
      " 'suffices to answer the question definitively. Therefore, this is a Case 2 '\n",
      " 'problem.\\n'\n",
      " '\\n'\n",
      " '[[2]]\\n')\n",
      "('In three piles there are $51, 49$, and $5$ stones, respectively. You can '\n",
      " 'combine any two piles into one pile or divide a pile consisting of an even '\n",
      " 'number of stones into two equal piles. Is it possible to get $105$ piles '\n",
      " 'with one stone in each?')\n",
      "'\\\\text{No}'\n",
      "('The problem asks whether it is possible to reach a certain configuration. '\n",
      " 'This is a yes/no question, and the answer provided gives a proof to show why '\n",
      " 'the answer is no. Therefore, this falls under Case 2.\\n'\n",
      " '\\n'\n",
      " '[[2]]\\n')\n",
      "('We are given $2021$ points on a plane, no three of which are collinear. '\n",
      " 'Among any $5$ of these points, at least $4$ lie on the same circle. Is it '\n",
      " 'necessarily true that at least $2020$ of the points lie on the same circle?')\n",
      "'\\\\text{Yes}'\n",
      "('The problem asks whether a statement is necessarily true. This matches the '\n",
      " 'criteria for Case 2, where a problem requires a proof to answer Yes/No. '\n",
      " 'Although the answer is given as \"Yes\", the core of the problem lies in '\n",
      " 'proving this statement, not in finding a direct answer.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{2}$\\n')\n",
      "('The positive integers from 1 to 100 are painted into three colors: 50 '\n",
      " 'integers are red, 25 integers are yellow and 25 integers are green. The red '\n",
      " 'and yellow integers can be divided into 25 triples such that each triple '\n",
      " 'includes two red integers and one yellow integer which is greater than one '\n",
      " 'of the red integers and smaller than another one. The same assertion is '\n",
      " 'valid for the red and green integers. Is it necessarily possible to divide '\n",
      " 'all the 100 integers into 25 quadruples so that each quadruple includes two '\n",
      " 'red integers, one yellow integer and one green integer such that the yellow '\n",
      " 'and the green integer lie between the red ones?')\n",
      "'\\\\text{Yes}'\n",
      "('The problem asks if it is necessarily possible to divide all 100 integers '\n",
      " 'into 25 quadruples satisfying certain conditions. The answer provided is '\n",
      " '\"Yes\".  This means the problem is asking for a proof of a statement. Thus, '\n",
      " 'this is Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('A set of points of the plane is called [i] obtuse-angled[/i] if every three '\n",
      " \"of it's points are not collinear and every triangle with vertices inside the \"\n",
      " 'set has one angle $ >91^o$. Is it correct that every finite [i] '\n",
      " 'obtuse-angled[/i] set can be extended to an infinite  [i]obtuse-angled[/i] '\n",
      " 'set?\\n'\n",
      " '\\n'\n",
      " '(UK)')\n",
      "'\\\\text{Yes}'\n",
      "('The problem asks whether every finite obtuse-angled set can be extended to '\n",
      " \"an infinite obtuse-angled set.  It's a yes/no question, implying a need for \"\n",
      " \"proof to justify the answer. Thus, it's a Case 2 problem.\\n\"\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is [[2]]\\n')\n",
      "('Baron Munchausen claims that he has drawn a polygon and chosen a point '\n",
      " 'inside the polygon in such a way that any line passing through the chosen '\n",
      " 'point divides the polygon into three polygons. Could the Baron’s claim be '\n",
      " 'correct?')\n",
      "'\\\\text{Yes}'\n",
      "('The problem asks whether it is possible for a polygon to be divided into '\n",
      " 'three polygons by any line passing through a chosen point inside the '\n",
      " 'polygon. The answer is a simple \"yes\".  This is a Case 2 problem disguised '\n",
      " 'as a Case 1 problem. The problem is essentially asking to prove/disprove the '\n",
      " 'existence of such a polygon. While the answer is a simple \"yes\", the '\n",
      " \"problem's core is about proving the existence, which requires a constructive \"\n",
      " 'proof (i.e., providing an example of such a polygon and point).\\n'\n",
      " '\\n'\n",
      " 'Final Answer: [[2]]\\n')\n",
      "('Given an infinite sequence of numbers $a_1, a_2, a_3,...$ . For each '\n",
      " 'positive integer $k$ there exists a positive integer $t = t(k)$ such that '\n",
      " '$a_k = a_{k+t} = a_{k+2t} =...$. Is this sequence necessarily periodic? That '\n",
      " 'is, does a positive integer $T$ exist such that $a_k = a_{k+T}$ for each '\n",
      " 'positive integer k?')\n",
      "'\\\\text{No}'\n",
      "('The problem asks a yes/no question about the properties of a sequence. The '\n",
      " 'answer provided is simply \"No,\" which implies a counterexample exists. This '\n",
      " 'suggests that the problem requires a proof to demonstrate why such a '\n",
      " \"sequence isn't necessarily periodic.  Therefore, it falls under Case 2.\\n\"\n",
      " '\\n'\n",
      " '[[2]]\\n')\n",
      "('For a positive integer $n$ we denote by $s(n)$ the sum of the digits of $n$. '\n",
      " 'Let $P(x)=x^n+a_{n-1}x^{n-1}+\\\\cdots+a_1x+a_0$ be a polynomial, where $n '\n",
      " '\\\\geqslant 2$ and $a_i$ is a positive integer for all $0 \\\\leqslant i '\n",
      " '\\\\leqslant n-1$. Could it be the case that, for all positive integers $k$, '\n",
      " '$s(k)$ and $s(P(k))$ have the same parity?')\n",
      "'\\\\text{There is no such polynomial.}'\n",
      "('The problem asks \"Could it be the case that...\" which implies a yes/no '\n",
      " 'question. The answer states \"There is no such polynomial\" which answers no. '\n",
      " 'The problem asks to prove/disprove if such polynomial exists, which falls '\n",
      " 'under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: [[2]]\\n')\n",
      "('Baron Munchausen presented a new theorem: if a polynomial $x^{n} - ax^{n-1} '\n",
      " '+ bx^{n-2}+ \\\\dots$ has $n$ positive integer roots then there exist $a$ '\n",
      " 'lines in the plane such that they have exactly $b$ intersection points. Is '\n",
      " 'the baron’s theorem true?')\n",
      "'\\\\text{True}'\n",
      "('The problem asks whether a theorem presented is true or not. Although the '\n",
      " 'answer provided states that it is true, the problem itself is asking to '\n",
      " 'prove or disprove a statement/theorem. Therefore, it falls under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: [[2]]\\n')\n",
      "('In a sequence $P_n$ of quadratic trinomials each trinomial, starting with '\n",
      " 'the third, is the sum of the two preceding trinomials. The first two '\n",
      " 'trinomials do not have common roots. Is it possible that $P_n$ has an '\n",
      " 'integral root for each $n$?')\n",
      "'\\\\text{no}'\n",
      "('The problem asks whether it is possible for each term in a sequence of '\n",
      " 'quadratic trinomials to have an integer root, given specific conditions on '\n",
      " 'the sequence. The answer provided is a simple \"no.\"  This indicates that the '\n",
      " 'problem is asking for a proof to validate the statement. Therefore, the '\n",
      " 'problem belongs to Case 2.\\n'\n",
      " '\\n'\n",
      " '[[2]]\\n')\n",
      "('Consider $2018$ pairwise crossing circles no three of which are concurrent. '\n",
      " 'These circles subdivide the plane into regions bounded by circular $edges$ '\n",
      " 'that meet at $vertices$. Notice that there are an even number of vertices on '\n",
      " 'each circle. Given the circle, alternately colour the vertices on that '\n",
      " 'circle red and blue. In doing so for each circle, every vertex is coloured '\n",
      " 'twice- once for each of the two circle that cross at that point. If the two '\n",
      " 'colours agree at a vertex, then it is assigned that colour; otherwise, it '\n",
      " 'becomes yellow. Show that, if some circle contains at least $2061$ yellow '\n",
      " 'points, then the vertices of some region are all yellow.')\n",
      "''\n",
      "('The problem asks to show that a statement is true. This clearly falls under '\n",
      " 'Case 2, since the problem is asking for a proof.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: [[2]]\\n')\n",
      "('There is a row of $100N$ sandwiches with ham. A boy and his cat play a game. '\n",
      " 'In one action the boy eats the first sandwich from any end of the row. In '\n",
      " 'one action the cat either eats the ham from one sandwich or does nothing. '\n",
      " 'The boy performs 100 actions in each of his turns, and the cat makes only 1 '\n",
      " 'action each turn; the boy starts first. The boy wins if the last sandwich he '\n",
      " 'eats contains ham. Is it true that he can win for any positive integer $N{}$ '\n",
      " 'no matter how the cat plays?')\n",
      "'\\\\text{No}'\n",
      "('The problem asks \"Is it true that he can win for any positive integer $N$ no '\n",
      " 'matter how the cat plays?\". This is equivalent to asking to prove or '\n",
      " 'disprove a statement. Thus, this is a Case 2 problem.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Consider an acute non-isosceles triangle. In a single step it is allowed to '\n",
      " 'cut any one of the available triangles into two triangles along its median. '\n",
      " 'Is it possible that after a finite number of cuttings all triangles will be '\n",
      " 'isosceles?')\n",
      "'\\\\text{No}'\n",
      "('The problem asks whether it is possible to cut an acute non-isosceles '\n",
      " 'triangle into isosceles triangles after a finite number of steps. The answer '\n",
      " 'provided is \"No\". This problem requires a proof to show why it is '\n",
      " 'impossible. Thus, it falls under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "(\"Anna's number is obtained by writing down $20$ consecutive positive \"\n",
      " \"integers, one after another in arbitrary order. Bob's number is obtained in \"\n",
      " 'the same way, but with $21$ consecutive positive integers. Can they obtain '\n",
      " 'the same number?')\n",
      "'\\\\text{Not equal}'\n",
      "('The problem asks \"Can they obtain the same number?\". This is a yes/no '\n",
      " 'question and the answer provided is \"Not equal\", implying no.  Therefore, '\n",
      " 'this problem falls under Case 2, which involves problems that require a '\n",
      " 'proof to answer Yes/No.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: [[2]]\\n')\n",
      "('A hunter and an invisible rabbit play a game in the Euclidean plane. The '\n",
      " \"rabbit's starting point, $A_0,$ and the hunter's starting point, $B_0$ are \"\n",
      " 'the same. After $n-1$ rounds of the game, the rabbit is at point $A_{n-1}$ '\n",
      " 'and the hunter is at point $B_{n-1}.$ In the $n^{\\\\text{th}}$ round of the '\n",
      " 'game, three things occur in order:\\n'\n",
      " '\\n'\n",
      " '[list=i]\\n'\n",
      " '[*]The rabbit moves invisibly to a point $A_n$ such that the distance '\n",
      " 'between $A_{n-1}$ and $A_n$ is exactly $1.$\\n'\n",
      " '\\n'\n",
      " '[*]A tracking device reports a point $P_n$ to the hunter. The only guarantee '\n",
      " 'provided by the tracking device to the hunter is that the distance between '\n",
      " '$P_n$ and $A_n$ is at most $1.$\\n'\n",
      " '\\n'\n",
      " '[*]The hunter moves visibly to a point $B_n$ such that the distance between '\n",
      " '$B_{n-1}$ and $B_n$ is exactly $1.$\\n'\n",
      " '[/list]\\n'\n",
      " 'Is it always possible, no matter how the rabbit moves, and no matter what '\n",
      " 'points are reported by the tracking device, for the hunter to choose her '\n",
      " 'moves so that after $10^9$ rounds, she can ensure that the distance between '\n",
      " 'her and the rabbit is at most $100?$\\n'\n",
      " '\\n'\n",
      " '[i]')\n",
      "'\\\\text{No}'\n",
      "('The problem asks whether it is always possible for the hunter to choose her '\n",
      " 'moves so that after $10^9$ rounds, the distance between her and the rabbit '\n",
      " 'is at most $100.$ This is a yes/no question, and the answer provided is '\n",
      " '\"No.\"  The problem asks to determine if such a strategy exists for the '\n",
      " 'hunter, not to explicitly find the strategy.  Thus, we are looking for a '\n",
      " 'proof that no such strategy exists. This falls under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Four integers are marked on a circle. On each step we simultaneously replace '\n",
      " 'each number by the difference between this number and next number on the '\n",
      " 'circle, moving in a clockwise direction; that is, the numbers $ a,b,c,d$ are '\n",
      " 'replaced by $ a\\\\minus{}b,b\\\\minus{}c,c\\\\minus{}d,d\\\\minus{}a.$ Is it '\n",
      " 'possible after 1996 such to have numbers $ a,b,c,d$ such the numbers $ '\n",
      " '|bc\\\\minus{}ad|, |ac \\\\minus{} bd|, |ab \\\\minus{} cd|$ are primes?')\n",
      "'\\\\text{No}'\n",
      "('The problem asks if it is possible to obtain a specific outcome after a '\n",
      " 'certain number of steps. While there might be some underlying proof or '\n",
      " 'reasoning involved in determining whether this is possible or not, the '\n",
      " 'problem itself is directly asking for a yes/no answer. Thus, the final '\n",
      " 'answer should be [[2]].\\n')\n",
      "('[i]Nyihaha[/i] and [i]Bruhaha[/i] are two neighbouring  islands, both having '\n",
      " '$n$ inhabitants. On island [i]Nyihaha[/i] every inhabitant is either a '\n",
      " 'Knight or a Knave. Knights always tell the truth and Knaves always lie. The '\n",
      " 'inhabitants of island [i]Bruhaha[/i] are normal people, who can choose to '\n",
      " 'tell the truth or lie. When a visitor arrives on any of the two islands, the '\n",
      " 'following ritual is performed: every inhabitant points randomly to another '\n",
      " 'inhabitant (indepently from each other with uniform distribution), and tells '\n",
      " '\"He is a Knight\" or \"He is a Knave\\'\". On sland [i]Nyihaha[/i], Knights have '\n",
      " 'to tell the truth and Knaves have to lie. On island [i]Bruhaha[/i] every '\n",
      " 'inhabitant tells the truth with probability $1/2$ independently from each '\n",
      " 'other. Sinbad arrives on island [i]Bruhaha[/i], but he does not know whether '\n",
      " 'he is on island [i]Nyihaha[/i] or island [i]Bruhaha[/i]. Let $p_n$ denote '\n",
      " 'the probability that after observing the ritual he can rule out being on '\n",
      " 'island [i]Nyihaha[/i]. Is it true that $p_n\\\\to 1$ if $n\\\\to\\\\infty$?')\n",
      "'p_n \\\\to 1 \\\\text{ as } n \\\\to \\\\infty'\n",
      "('The problem asks \"Is it true that $p_n\\\\to 1$ if $n\\\\to\\\\infty$?\". This is a '\n",
      " 'yes/no question, which typically falls under Case 2. However, since the '\n",
      " 'provided answer is a definitive \"p_n \\\\to 1 \\\\text{ as } n \\\\to \\\\infty\", it '\n",
      " 'implies that the answer to the yes/no question is \"yes,\" and this serves as '\n",
      " 'a direct answer. Therefore, this problem belongs to Case 1.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{1}$\\n')\n",
      "('Does the set $\\\\{1,2,3,...,3000\\\\}$ contain a subset $ A$ consisting of 2000 '\n",
      " 'numbers that $x\\\\in A$ implies $2x \\\\notin A$ ?!!  :?:')\n",
      "'\\\\text{No}'\n",
      "('The problem asks whether there exists a subset with a given property. This '\n",
      " 'is a yes/no question, which implies a proof is required. Thus, this is Case '\n",
      " '2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is [[2]]\\n')\n",
      "('Can there be drawn on a circle of radius $1$ a number of $1975$ distinct '\n",
      " 'points, so that the distance (measured on the chord) between any two points '\n",
      " '(from the considered points) is a rational number?')\n",
      "'\\\\text{yes}'\n",
      "(('Let $a_0$, $a_1$, $a_2$, ... be an infinite sequence of real numbers ''The problem asks \"Can there be drawn...\" which implies a yes/no answer.  The '\n",
      " \n",
      " 'provided answer is simply \"yes\". This strongly suggests the problem requires ''satisfying the equation $a_n=\\\\left|a_{n+1}-a_{n+2}\\\\right|$ for all $n\\\\geq '\n",
      " \n",
      " 'a proof to justify the \"yes\" answer.  Therefore, it falls under Case 2.\\n''0$, where $a_0$ and $a_1$ are two different positive reals.\\n'\n",
      " \n",
      " '\\n''\\n'\n",
      " \n",
      " '[[2]]\\n''Can this sequence $a_0$, $a_1$, $a_2$, ... be bounded?\\n')\n",
      " \n",
      "'\\n'\n",
      " '[i]')\n",
      "'\\\\text{No}'\n",
      "('The problem asks \"Can this sequence be bounded?\". This is equivalent to '\n",
      " 'asking to prove or disprove the existence of a bounded sequence satisfying '\n",
      " 'the given conditions.  Thus, it is a proof-based problem.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{2}$\\n')\n",
      "('Are there any triples $(a,b,c)$ of positive integers such that '\n",
      " '$(a-2)(b-2)(c-2)+12$ is a prime number that properly divides the positive '\n",
      " 'number $a^2+b^2+c^2+abc-2017$?')\n",
      "'$\\\\text{no such }(a,b,c)$'\n",
      "('The problem asks whether there exist positive integers $a, b, c$ satisfying '\n",
      " 'a given condition. This is a yes/no question, which typically falls under '\n",
      " 'Case 2. Although the answer provided is \"no,\" the problem itself requires a '\n",
      " 'proof to justify this answer. Thus, this corresponds to proving there are no '\n",
      " 'such triples $(a, b, c)$.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: [[2]]\\n')\n",
      "('Is it possible to cover a $13\\\\times 13$ chessboard with forty-two pieces of '\n",
      " 'dimensions $4\\\\times 1$ such that only the central square of the chessboard '\n",
      " 'remains uncovered?')\n",
      "'\\\\text{No}'\n",
      "('The problem asks a yes/no question, which suggests it could be Case 2. '\n",
      " 'However, the answer provided is simply \"No,\" without any accompanying proof '\n",
      " 'or reasoning. Thus, while the problem itself might require a proof to answer '\n",
      " 'definitively, the provided information only gives the answer without '\n",
      " 'justification. This makes it difficult to classify definitively as Case 2. '\n",
      " 'The presence of a clear answer (\"No\") suggests it could fall under Case 1. '\n",
      " 'However, the problem is phrased as a question of possibility. Therefore, '\n",
      " 'without proof, it is classified as a Case 2 problem.\\n'\n",
      " '\\n'\n",
      " '[[2]]\\n')\n",
      "('Do there exist 100 positive distinct integers such that a cube of one of '\n",
      " 'them equals the sum of the cubes of all the others?')\n",
      "'$\\\\text{Yes}$'\n",
      "('The problem asks whether there exist 100 distinct positive integers such '\n",
      " 'that the cube of one equals the sum of the cubes of the others. This is a '\n",
      " 'yes/no question, thus requiring a proof to show the existence or '\n",
      " 'non-existence of such integers. Therefore, it falls under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Let $ABC$ be a triangle with $AB < AC$, incenter $I$, and $A$ excenter '\n",
      " '$I_{A}$. The incircle meets $BC$ at $D$. Define $E = AD\\\\cap BI_{A}$, $F = '\n",
      " 'AD\\\\cap CI_{A}$. Show that the circumcircle of $\\\\triangle AID$ and '\n",
      " '$\\\\triangle I_{A}EF$ are tangent to each other')\n",
      "('\\\\text{The circumcircles of } \\\\triangle AID \\\\text{ and } \\\\triangle I_AEF '\n",
      " '\\\\text{ are tangent to each other.}')\n",
      "('The problem asks to show/prove that two circumcircles are tangent to each '\n",
      " 'other. This is a problem that requires a proof to show that a statement is '\n",
      " 'true. Thus, this is Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: [[2]]\\n')\n",
      "('Can the product of $2$ consecutive natural numbers equal the product of $2$ '\n",
      " 'consecutive even natural numbers?\\n'\n",
      " '(natural means positive integers)')\n",
      "'\\\\text{No.}'\n",
      "('The problem asks \"Can the product ...?\". This is equivalent to asking '\n",
      " 'whether there exists a solution, which implies a yes/no answer. Thus, it '\n",
      " 'falls under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Initially, the number $1$ and a non-integral number $x$ are written on a '\n",
      " 'blackboard. In each step, we can choose two numbers on the blackboard, not '\n",
      " 'necessarily different, and write their sum or their difference on the '\n",
      " 'blackboard. We can also choose a non-zero number of the blackboard and write '\n",
      " 'its reciprocal on the blackboard. Is it possible to write $x^2$ on the '\n",
      " 'blackboard in a finite number of moves?')\n",
      "'\\\\text{No}'\n",
      "('The problem asks whether it is possible to write $x^2$ on the blackboard in '\n",
      " 'a finite number of moves, given certain allowed operations. The answer '\n",
      " 'provided is \"No.\" Since the problem poses a yes/no question and requires a '\n",
      " 'proof to justify the answer, it falls under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Two players play the following game. The first player starts by writing '\n",
      " 'either $0$ or $1$ and then, on his every move, chooses either $0$ or $1$ and '\n",
      " 'writes it to the right of the existing digits until there are $1999$ digits. '\n",
      " 'Each time the first player puts down a digit (except the first one) , the '\n",
      " 'second player chooses two digits among those already written and swaps them. '\n",
      " 'Can the second player guarantee that after his last move the line of digits '\n",
      " 'will be symmetrical about the middle digit? \\n'\n",
      " '\\n'\n",
      " '(I Izmestiev)')\n",
      "'\\\\text{Yes}'\n",
      "('The problem asks whether the second player can guarantee a symmetrical '\n",
      " 'arrangement of digits. The answer provided is a simple \"Yes\".  This implies '\n",
      " 'the problem is asking for a proof of the existence of a strategy. Therefore, '\n",
      " 'this is a Case 2 problem.\\n'\n",
      " '\\n'\n",
      " '[[2]]\\n')\n",
      "('Given three letters $X, Y, Z$, we can construct letter sequences '\n",
      " 'arbitrarily, such as $XZ, ZZYXYY, XXYZX$, etc. For any given sequence, we '\n",
      " 'can perform following operations:\\n'\n",
      " '\\n'\n",
      " '$T_1$: If the right-most letter is $Y$, then we can add $YZ$ after it, for '\n",
      " 'example, $T_1(XYZXXY) =\\n'\n",
      " '(XYZXXYYZ).$\\n'\n",
      " '\\n'\n",
      " '$T_2$: If The sequence contains $YYY$, we can replace them by $Z$, for '\n",
      " 'example, $T_2(XXYYZYYYX) =\\n'\n",
      " '(XXYYZZX).$\\n'\n",
      " '\\n'\n",
      " '$T_3$: We can replace $Xp$ ($p$ is any sub-sequence) by $XpX$, for example, '\n",
      " '$T_3(XXYZ) = (XXYZX).$\\n'\n",
      " '\\n'\n",
      " '$T_4$: In a sequence containing one or more $Z$, we can replace the first '\n",
      " '$Z$ by $XY$, for example,\\n'\n",
      " '$T_4(XXYYZZX) = (XXYYXYZX).$\\n'\n",
      " '\\n'\n",
      " '$T_5$: We can replace any of $XX, YY, ZZ$ by $X$, for example, $T_5(ZZYXYY) '\n",
      " '= (XYXX)$ or $(XYXYY)$ or $(ZZYXX).$\\n'\n",
      " '\\n'\n",
      " 'Using above operations, can we get $XYZZ$ from $XYZ \\\\ ?$')\n",
      "'\\\\text{no}'\n",
      "('The problem asks whether it is possible to obtain the sequence XYZZ from XYZ '\n",
      " 'using a set of given operations.  This is a yes/no question, and the '\n",
      " 'provided answer is \"no.\"  The problem statement doesn\\'t explicitly ask for '\n",
      " 'a proof, but the nature of the problem strongly implies that justification '\n",
      " 'is required, making this a proof-based problem.\\n'\n",
      " '\\n'\n",
      " '[[2]]\\n')\n",
      "('There is a population $P$ of $10000$ bacteria, some of which are friends '\n",
      " '(friendship is mutual),\\n'\n",
      " 'so that each bacterion has at least one friend and if we wish to assign to '\n",
      " 'each bacterion a coloured\\n'\n",
      " 'membrane so that no two friends have the same colour, then there is a way to '\n",
      " 'do it with $2021$\\n'\n",
      " 'colours, but not with $2020$ or less.\\n'\n",
      " 'Two friends $A$ and $B$ can decide to merge in which case they become a '\n",
      " 'single bacterion whose\\n'\n",
      " 'friends are precisely the union of friends of $A$ and $B$. (Merging is not '\n",
      " 'allowed if $A$ and $B$ are\\n'\n",
      " 'not friends.) It turns out that no matter how we perform one merge or two '\n",
      " 'consecutive merges,\\n'\n",
      " 'in the resulting population it would be possible to assign $2020$ colours or '\n",
      " 'less so that no two\\n'\n",
      " 'friends have the same colour. Is it true that in any such population $P$ '\n",
      " 'every bacterium has at\\n'\n",
      " 'least $2021$ friends?')\n",
      "'\\\\text{True}'\n",
      "('The problem asks if a statement is true, which implies a yes/no answer. This '\n",
      " \"falls under Case 2, as it's asking for a proof related to a statement about \"\n",
      " 'a population of bacteria and their friendships. While the answer is provided '\n",
      " 'as \"True,\" the core of the problem is to prove this statement, making it a '\n",
      " 'proof-based problem.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: [[2]]\\n')\n",
      "('Can the positive integers be partitioned into $12$ subsets such that for '\n",
      " 'each positive integer $k$, the numbers $k, 2k,\\\\ldots,12k$ belong to '\n",
      " 'different subsets?')\n",
      "'\\\\text{No}'\n",
      "('The problem asks a yes/no question and the answer provided is simply \"No\".  '\n",
      " 'This strongly suggests the problem requires a proof to justify the \"no\" '\n",
      " 'answer. Therefore, this falls under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Given $ n$ countries with three representatives each, $ m$ committees $ '\n",
      " 'A(1),A(2), \\\\ldots, A(m)$ are called a cycle if\\n'\n",
      " '\\n'\n",
      " '[i](i)[/i] each committee has $ n$ members, one from each country;\\n'\n",
      " '[i](ii)[/i] no two committees have the same membership;\\n'\n",
      " '[i](iii)[/i] for $ i \\\\equal{} 1, 2, \\\\ldots,m$, committee $ A(i)$ and '\n",
      " 'committee $ A(i \\\\plus{} 1)$ have no member in common, where $ A(m \\\\plus{} '\n",
      " '1)$ denotes $ A(1);$\\n'\n",
      " '[i](iv)[/i] if $ 1 < |i \\\\minus{} j| < m \\\\minus{} 1,$ then committees $ '\n",
      " 'A(i)$ and $ A(j)$ have at least one member in common.\\n'\n",
      " '\\n'\n",
      " 'Is it possible to have a cycle of 1990 committees with 11 countries?')\n",
      "'\\\\text{yes}'\n",
      "('The problem asks whether it is possible to have a cycle of 1990 committees '\n",
      " 'with 11 countries. The answer provided is yes. Since the problem presents a '\n",
      " 'yes/no question and requires a construction to prove the existence, this '\n",
      " 'falls under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Let $p_n$ denote the $n^{\\\\text{th}}$ prime number and define $a_n=\\\\lfloor '\n",
      " 'p_n\\\\nu\\\\rfloor$ for all positive integers $n$ where $\\\\nu$ is a positive '\n",
      " 'irrational number. Is it possible that there exist only finitely many $k$ '\n",
      " 'such that $\\\\binom{2a_k}{a_k}$ is divisible by $p_i^{10}$ for all '\n",
      " '$i=1,2,\\\\ldots,2020?$')\n",
      "'\\\\text{No}'\n",
      "('The problem asks a yes/no question about the existence of finitely many $k$ '\n",
      " 'satisfying a given divisibility condition. While an answer (\"No\") is '\n",
      " 'provided, the core of the problem lies in proving or disproving the '\n",
      " 'statement. Thus, this falls under Case 2.\\n'\n",
      " '\\n'\n",
      " '[[2]]\\n')\n",
      "('Every positive integer greater than $1000$ is colored in red or blue, such '\n",
      " 'that the product of any two distinct red numbers is blue. Is it possible to '\n",
      " 'happen that no two blue numbers have difference $1$?')\n",
      "'\\\\text{No}'\n",
      "('The problem asks whether a coloring with a given property is possible. This '\n",
      " 'is a yes/no question, and the answer provided is \"no.\"  Therefore, this '\n",
      " 'problem falls under Case 2.\\n'\n",
      " '\\n'\n",
      " '[[2]]\\n')\n",
      "Exception:  503 The service is currently unavailable.\n",
      "Exception:  503 The service is currently unavailable.\n",
      "Exception:  503 The service is currently unavailable.\n",
      "Exception:  503 The service is currently unavailable.\n",
      "('Do there exist positive integers $a_1<a_2<\\\\ldots<a_{100}$ such that for '\n",
      " '$2\\\\le k\\\\le100$, the least common multiple of $a_{k-1}$ and $a_k$ is '\n",
      " 'greater than the least common multiple of $a_k$ and $a_{k+1}$?')\n",
      "'\\\\text{Yes}'\n",
      "('The problem asks whether such integers exist. Although the answer provided '\n",
      " 'is a simple \"yes,\" the problem itself requires a proof to demonstrate the '\n",
      " 'existence of these integers. Thus, this falls under Case 2.\\n'\n",
      " '\\n'\n",
      " '[[2]]\\n')\n",
      "('From a set of integers $\\\\{1,...,100\\\\}$, $k$ integers were deleted. Is it '\n",
      " 'always possible to choose $k$ distinct integers from the remaining set such '\n",
      " 'that their sum is $100$ if\\n'\n",
      " '\\n'\n",
      " '[b](a) $k=9$?[/b]\\n'\n",
      " '[b](b) $k=8$?[/b]')\n",
      "'\\\\text{No}\\\\text{Yes}'\n",
      "('The problem asks \"Is it always possible to choose $k$ distinct integers '\n",
      " '...\".  This is a yes/no question, which suggests a proof might be required. '\n",
      " 'The answer provided is simply \"NoYes\".  This further reinforces that the '\n",
      " 'problem requires a proof to show why the answer is \"No\" for part (a) and '\n",
      " '\"Yes\" for part (b). This makes it a Case 2 problem.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "Exception:  503 The service is currently unavailable.\n",
      "('Let us say that a pair of distinct positive integers is nice if their '\n",
      " 'arithmetic mean and their geometric mean are both integer. Is it true that '\n",
      " 'for each nice pair there is another nice pair with the same arithmetic mean? '\n",
      " '(The pairs $(a, b)$ and $(b, a)$ are considered to be the same pair.)')\n",
      "'\\\\text{Yes}'\n",
      "('The problem asks to prove/disprove a statement. The problem is clearly '\n",
      " 'asking if it\\'s true or not that \"for each nice pair there is another nice '\n",
      " 'pair with the same arithmetic mean\". Thus, it falls under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Four positive integers $x,y,z$ and $t$ satisfy the relations\\n'\n",
      " '\\\\[ xy - zt = x + y = z + t. \\\\]\\n'\n",
      " 'Is it possible that both $xy$ and $zt$ are perfect squares?')\n",
      "'No'\n",
      "('The problem asks a yes/no question, \"Is it possible that both $xy$ and $zt$ '\n",
      " 'are perfect squares?\". This clearly falls under Case 2 since the answer '\n",
      " 'requires a proof to answer the question.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Let $ABC$ be a triangle with $\\\\angle ABC$ obtuse. The [i]$A$-excircle[/i] '\n",
      " 'is a circle in the exterior of $\\\\triangle ABC$ that is tangent to side $BC$ '\n",
      " 'of the triangle and tangent to the extensions of the other two sides. Let '\n",
      " '$E$, $F$ be the feet of the altitudes from $B$ and $C$ to lines $AC$ and '\n",
      " '$AB$, respectively. Can line $EF$ be tangent to the $A$-excircle?')\n",
      "'\\\\text{Line } EF \\\\text{ cannot be tangent to the } A\\\\text{-excircle.}'\n",
      "('The problem asks \"Can line EF be tangent to the $A$-excircle?\". This is '\n",
      " 'equivalent to asking to prove or disprove that line $EF$ can be tangent to '\n",
      " 'the $A$-excircle. Thus, this problem falls under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('A positive integer is called [i]downhill[/i] if the digits in its decimal '\n",
      " 'representation form a nonstrictly decreasing sequence from left to right. '\n",
      " 'Suppose that a polynomial $P(x)$ with rational coefficients takes on an '\n",
      " 'integer value for each downhill positive integer $x$. Is it necessarily true '\n",
      " 'that $P(x)$ takes on an integer value for each integer $x$?')\n",
      "'\\\\text{No}'\n",
      "('The problem asks to determine whether a polynomial with certain properties '\n",
      " '*necessarily* takes on an integer value for each integer $x$. This is '\n",
      " 'equivalent to asking whether or not a statement is true, so a proof is '\n",
      " 'required. Therefore, this problem falls under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('For $ x \\\\in (0, 1)$ let $ y \\\\in (0, 1)$ be the number whose $ n$-th digit '\n",
      " 'after the decimal point is the $ 2^{n}$-th digit after the decimal point of '\n",
      " '$ x$. Show that if $ x$ is rational then so is $ y$.\\n'\n",
      " '\\n'\n",
      " '[i]')\n",
      "'\\\\text{If } x \\\\text{ is rational, then } y \\\\text{ is rational.}'\n",
      "('The problem asks to show that if $x$ is rational, then $y$ is rational. This '\n",
      " 'is a proof-based problem, falling under Case 2. Although the answer states '\n",
      " '\"If x is rational, then y is rational\", the core of the problem lies in '\n",
      " 'proving this statement.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "Exception:  503 The service is currently unavailable.\n",
      "Exception:  503 The service is currently unavailable.\n",
      "Exception:  503 The service is currently unavailable.\n",
      "('A sequence of real numbers $x_1,x_2,\\\\ldots ,x_n$ is given such that '\n",
      " '$x_{i+1}=x_i+\\\\frac{1}{30000}\\\\sqrt{1-x_i^2},\\\\ i=1,2,\\\\ldots ,$ and '\n",
      " '$x_1=0$. Can $n$ be equal to $50000$ if $x_n<1$?')\n",
      "'\\\\text{No}'\n",
      "('The problem asks whether $n$ can be a specific value ($50000$) given a '\n",
      " 'condition ($x_n < 1$). Although it is a mathematical problem and the answer '\n",
      " 'provided is \"no\", the question posed requires a proof to definitively answer '\n",
      " 'yes or no. Therefore, it falls under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Let $ L$ denote the set of all lattice points of the plane (points with '\n",
      " 'integral coordinates). Show that for any three points $ A,B,C$ of $ L$ there '\n",
      " 'is a fourth point $ D,$ different from $ A,B,C,$ such that the interiors of '\n",
      " 'the segments $ AD,BD,CD$ contain no points of $ L.$ Is the statement true if '\n",
      " 'one considers four points of $ L$ instead of three?')\n",
      "'\\\\text{Yes}'\n",
      "('The problem asks to show that a fourth point D exists with a specific '\n",
      " 'property related to lattice points. Although phrased as \"show that,\" the '\n",
      " 'core task is to demonstrate the existence of such a point, making it a '\n",
      " 'proof-based problem. The answer is a simple \"Yes,\" further confirming its '\n",
      " 'nature as a proof-based question. Thus, this falls under Case 2.\\n'\n",
      " '\\n'\n",
      " '[[2]]\\n')\n",
      "('[i]Superchess[/i] is played on on a $12 \\\\times 12$ board, and it uses '\n",
      " '[i]superknights[/i], which move between opposite corner cells of any '\n",
      " '$3\\\\times4$ subboard. Is it possible for a [i]superknight[/i] to visit every '\n",
      " 'other cell of a superchessboard exactly once and return to its starting cell '\n",
      " '?')\n",
      "'No'\n",
      "('The problem asks whether it is possible for a superknight to visit every '\n",
      " 'other cell of a superchessboard exactly once and return to its starting '\n",
      " 'cell. This is a yes/no question and the answer provided is \"No\". This '\n",
      " 'problem falls under Case 2 because it requires a proof to show why it is not '\n",
      " 'possible.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Let $a_0$ be a positive integer and $a_n=5a_{n-1}+4$ for all $n\\\\ge 1$. Can '\n",
      " '$a_0$ be chosen so that $a_{54}$ is a multiple of $2013$?')\n",
      "'\\\\text{No}'\n",
      "('The problem asks whether there exists a choice of $a_0$ such that $a_{54}$ '\n",
      " 'is a multiple of $2013$. This is equivalent to asking if there exists a '\n",
      " 'solution to the congruence $a_{54} \\\\equiv 0 \\\\pmod{2013}$. Thus, the '\n",
      " 'problem asks for a Yes/No answer, which usually implies a proof. Thus, this '\n",
      " 'is Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Consider two positive integers $a$ and $b$ such that $a^{n+1} + b^{n+1}$ is '\n",
      " 'divisible by $a^n + b^n$ for infi\\x0c'\n",
      " 'nitely many positive integers $n$. Is it necessarily true that $a = b$?\\n'\n",
      " '\\n'\n",
      " '(Boris Frenkin)')\n",
      "'$   \\\\text { No }  $'\n",
      "('The problem asks to prove or disprove a statement. Specifically, it asks \"Is '\n",
      " 'it necessarily true that $a=b$?\". Thus, it falls under Case 2.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: The final answer is $\\\\boxed{[[2]]}$\\n')\n",
      "('Suppose that $k$ is a positive integer. A bijective map $f : Z \\\\to Z$ is '\n",
      " 'said to be $k$-[i]jumpy [/i] if $|f(z) - z| \\\\le k$ for all integers $z$.\\n'\n",
      " 'Is it that case that for every $k$, each $k$-jumpy map is a composition of '\n",
      " '$1$-jumpy maps?\\n'\n",
      " '[i]It is well known that this is the case when the support of the map is '\n",
      " 'finite.[/i]')\n",
      "'\\\\text{Yes}'\n",
      "('The problem asks to determine whether for every positive integer \\\\(k\\\\), '\n",
      " 'each \\\\(k\\\\)-jumpy map is a composition of \\\\(1\\\\)-jumpy maps, given that a '\n",
      " '\\\\(k\\\\)-jumpy map is a bijective map \\\\(f: \\\\mathbb{Z} \\\\to \\\\mathbb{Z}\\\\) '\n",
      " 'such that \\\\(|f(z) - z| \\\\le k\\\\) for all integers \\\\(z\\\\).  Since the '\n",
      " 'problem is a yes/no question and requires a proof, this is a Case 2 '\n",
      " 'problem.\\n'\n",
      " '\\n'\n",
      " 'Final Answer: [[2]]\\n')\n",
      "Exception:  503 The service is currently unavailable.\n",
      "Exception:  503 The service is currently unavailable.\n",
      "Exception:  503 The service is currently unavailable.\n",
      "Exception:  503 The service is currently unavailable.\n",
      "Exception:  503 The service is currently unavailable.\n",
      "Exception:  503 The service is currently unavailable.\n",
      "Exception:  503 The service is currently unavailable.\n",
      "Exception:  503 The service is currently unavailable.\n",
      "Exception:  503 The service is currently unavailable.\n",
      "Exception:  503 The service is currently unavailable.\n",
      "Exception:  503 The service is currently unavailable.\n",
      "Exception:  503 The service is currently unavailable.\n",
      "Exception:  503 The service is currently unavailable.\n",
      "Exception:  503 The service is currently unavailable.\n",
      "Exception:  503 The service is currently unavailable.\n",
      "Exception:  503 The service is currently unavailable.\n",
      "Exception:  503 The service is currently unavailable.\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Processing entries: 100%|██████████| 4422/4422 [00:00<00:00, 150473.49it/s]\n"
     ]
    }
   ],
   "source": [
    "# Filter for non-proof problems.\n",
    "from concurrent.futures import ProcessPoolExecutor, as_completed\n",
    "from copy import deepcopy\n",
    "from pprint import pprint\n",
    "\n",
    "from tqdm import tqdm\n",
    "\n",
    "from rllm.system_prompts import FILTER_PROOF_PROMPT\n",
    "from rllm.utils import call_gemini_llm\n",
    "\n",
    "\n",
    "def filter_proofs(idx, entry):\n",
    "    # 1) Get the problem text\n",
    "    problem_text = entry[\"problem\"]\n",
    "    solution_text = entry[\"answer\"]\n",
    "    # 2) Call Gemini LLM\n",
    "    output_str = call_gemini_llm(f\"Problem: {problem_text} \\n\\n Answer: {solution_text}\", system_prompt=FILTER_PROOF_PROMPT, temperature=0.8, n=4)\n",
    "    if not output_str:\n",
    "        return idx, entry\n",
    "    for output in output_str:\n",
    "        if \"[[1]]\" in output:\n",
    "            return idx, entry\n",
    "    pprint(problem_text)\n",
    "    pprint(solution_text)\n",
    "    pprint(output_str[0])\n",
    "    return idx, {}\n",
    "\n",
    "\n",
    "data = deepcopy(omni)\n",
    "\n",
    "with ProcessPoolExecutor(max_workers=32) as executor:\n",
    "    # 1) Submit all jobs to the executor\n",
    "    futures = [executor.submit(filter_proofs, f_idx, entry) for f_idx, entry in enumerate(data)]\n",
    "\n",
    "# 2) Process them as they complete, using tqdm for a progress bar\n",
    "for future in tqdm(as_completed(futures), total=len(futures), desc=\"Processing entries\"):\n",
    "    # Get the result for each completed future\n",
    "    idx, result = future.result()\n",
    "    data[idx] = result\n",
    "data = [d for d in data if d]\n",
    "# Save final list as json\n",
    "with open(\"omni_math.json\", \"w\") as f:\n",
    "    json.dump(data, f, indent=2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4158"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "len(data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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